Virtual University of Pakistan Lecture No. 21 of the course on Statistics and Probability by Miss Saleha Naghmi Habibullah

IN THE LAST LECTURE, YOU LEARNT

•Application of Addition Theorem •Conditional Probability •Multiplication Theorem

TOPICS FOR TODAY •Independent and Dependent Events •Multiplication Theorem of Probability for Independent Events •Marginal Probability

Before we proceed the concept of independent versus dependent events, let us review the Addition and Multiplication Theorems of Probability that were discussed in the last lecture. To this end, let us consider an interesting example that illustrates the application of both of these theorems in one problem: EXAMPLE A bag contains 10 white and 3 black balls. Another bag contains 3 white and 5 black balls. Two balls are transferred from first bag and placed in the second, and then one ball is taken from the latter. What is the probability that it is a white ball?

In the beginning experiment, we have:

of

the

Colour of Ball White

No. of Balls in Bag A 10

No. of Balls in Bag B 3

Black

3

5

Total

13

8

Let A represent the event that 2 balls are drawn from the first bag and transferred to the second bag. Then A can occur in the following three mutually exclusive ways: A1 = 2 white balls are transferred to the second bag. A2 = 1 white ball and 1 black ball are transferred to the second bag. A3 = 2 black balls are transferred to the second bag.

Then, the total number of ways in which 2 balls can be drawn out 13 of a total of 13 balls is . 2

And, the total number of ways in which 2 white balls can be drawn 10 out of 10 white balls is . 2

Thus, the probability that two white balls are selected from the first bag containing 13 balls (in order to transfer to the second bag) is

10 13 45 P( A1 ) = ÷ = , 2 2 78

Similarly, the probability that one white ball and one black ball are selected from the first bag containing 13 balls (in order to transfer to the second bag) is

10 3 13 30 P( A 2 ) = ÷ = , 1 1 2 78

And, the probability that two black balls are selected from the first bag containing 13 balls (in order to transfer to the second bag) is

3 13 3 P( A 3 ) = ÷ = . 2 2 78

AFTER having transferred 2 balls from the first bag, the second bag contains i)

5 white and 5 black balls (if 2 white balls are transferred) Colour of Ball White

No. of Balls in Bag A 10 â€“ 2 = 8

No. of Balls in Bag B 3+2=5

Black

3

5

Total

13 â€“ 2 = 11

8 + 2 = 10

Hence: P(W/A1) = 5/10

ii) 4 white and 6 black balls (if 1 white and 1 black balls are transferred) Colour of Ball White

No. of Balls in Bag A 10 – 1 = 7

No. of Balls in Bag B 3+1=4

Black

3–1=2

5+1=4

Total

13 – 2 = 11

8 + 2 = 10

Hence: P(W/A2) = 4/10

iii) 3 white and 7 black balls (if 2 black balls are transferred) Colour of Ball White

No. of Balls in Bag A 10

No. of Balls in Bag B 3

Black

3–2=1

5+2=7

Total

13 – 2 = 11

8 + 2 = 10

Hence: P(W/A3) = 3/10

Let W represent the event that the WHITE ball is drawn from the second bag after having transferred 2 balls from the first bag. Then

P(W) = P(A1∩W) + P(A2∩W) + P(A3∩W)

Now P(A1 ∩ W) = P(A1)P(W/A1) = 45/78 × 5/10 = 15/52 P(A2 ∩ W) = P(A2)P(W/A2) = 30/78 × 4/10 = 2/13, and P(A3 ∩ W) = P(A3)P(W/A3) = 3/78 × 3/10 = 3/260. Hence the required probability is P(W) = P(A1∩W) + P(A2∩W) + P(A3∩W) = 15/52 + 2/13 + 3/260 = 59/130 = 0.45

INDEPENDENT EVENTS Two events A and B in the same sample space S, are defined to be independent (or statistically independent) if the probability that one event occurs, is not affected by whether the other event has or has not occurred, that is P(A/B) = P(A) and P(B/A) = P(B).

It then follows that two events A and B are independent if and only if P(A âˆŠ B) = P(A) P(B) and this is known as the special case of the Multiplication Theorem of Probability.

RATIONALE According to multiplication theorem probability, we have:

the of

P(A ∩ B) = P(A) . P(B/A) Putting P(B/A) = P(B), we obtain P(A ∩ B) = P(A) P(B)

The events A and B are defined to be DEPENDENT if P(A∩B) ≠ P(A) × P(B). This means that the occurrence of one of the events in some way affects the probability of the occurrence of the other event.

Speaking of independent events, it is to be emphasized that two events that are independent, can NEVER be mutually exclusive.

EXAMPLE: Two fair dice, one red and one green, are thrown. Let A denote the event that the red die shows an even number and let B denote the event that the green die shows a 5 or a 6. Show that the events A and B are independent.

The sample space S is represented by the following 36 outcomes: S = {(1, 1), (1, 2), (1, 3), (1, 5), (1, 6); (2, 1), (2, 2), (2, 3), (2, 5), (2, 6); (3, 1), (3, 2), (3, 3), (3, 5), (3, 6); (4, 1), (4, 2), (4, 3), (4, 5), (4, 6); (5, 1), (5, 2), (5, 3), (5, 5), (5, 6); (6, 1), (6, 2), (6, 3), (6, 5), (6, 6) }

Since A represents the event that red die shows an even number, and B represents the event that green die shows a 5 or a 6, Therefore A âˆŠ B represents the event that red die shows an even number and green die shows a 5 or a 6.

Since A represents the event that red die shows an even number, hence P(A) = 3/6. Similarly, since B represents the event that green die shows a 5 or a 6, hence P(B) = 2/6.

Now, in order to compute the probability of the joint event A âˆŠ B, the first point to note is that, in all, there are 36 possible outcomes when we throw the two dice together, i.e.

S = {(1, 1), (1, 2), (1, 3), (1, 5), (1, 6); (2, 1), (2, 2), (2, 3), (2, 5), (2, 6); (3, 1), (3, 2), (3, 3), (3, 5), (3, 6); (4, 1), (4, 2), (4, 3), (4, 5), (4, 6); (5, 1), (5, 2), (5, 3), (5, 5), (5, 6); (6, 1), (6, 2), (6, 3), (6, 5), (6, 6) }

The joint event A âˆŠ B contains only 6 outcomes out of the 36 possible outcomes. These are (2, 5), (4, 5), (6, 5), (2, 6), (4, 6), and (6, 6). and

P(A âˆŠ B) = 6/36.

Now P(A) P(B) = 3/6 Ă— 2/6 = 6/36 = P(A âˆŠ B). Therefore the events A and B are independent.

Three events A, B and C all defined on the same sample space, are said to be mutually independent if they satisfy the following conditions:

i) They are pairwise independent, i.e. P(A ∩ B) = P(A) P(B) P(A ∩ C) = P(A) P(C) P(B ∩ C) = P(B) P(C) ii) They are mutually independent, i.e. P(A ∩ B ∩C) = P(A) P(B) P(C)

Example Suppose that a coin, a red die and a blue die are tossed together. Let A denote the event that the coin shows head, let B denote the event the red die shows an even number, and C the event that the blue die shows a six.

Solution First of all, let us consider the three events A, B and C separately. Assuming that the coin as well as the two dice are fair, we have P(A) = 1/2 P(B) = 3/6 P(C) = 1/6

1) When the coin and the red die are tossed together, our sample space consists of twelve outcomes i.e. twelve ordered pairs which are: (H , 1), (H , 2), (H , 3), (H , 4), (H , 5), (H , 6), (T , 1), (T , 2), (T , 3), (T , 4), (T , 5), (T , 6)

Out of these, three ordered pairs favour the event A ∩ B i.e. the event that we obtain a head on the coin and an even number on the red die. These are (H , 2), (H , 4) and (H , 6). Hence, the probability of A ∩ B is 3/12 i.e. we obtain P(A ∩ B) = 3/12

2) When the coin and the blue die are tossed together, our sample space again consists of twelve outcomes i.e. twelve ordered pairs which are: (H , 1), (H , 2), (H , 3), (H , 4), (H , 5), (H , 6), (T , 1), (T , 2), (T , 3), (T , 4), (T , 5), (T , 6) i.e. we obtain P(A âˆŠ C) = 1/12

3) When the red die and the blue die are tossed together, our sample space consists of thirty-six outcomes i.e. 36 ordered pairs which are: (1, 1), (1, 2), (1, 3), (1, 5), (1, 6); (2, 1), (2, 2), (2, 3), (2, 5), (2, 6); (3, 1), (3, 2), (3, 3), (3, 5), (3, 6); (4, 1), (4, 2), (4, 3), (4, 5), (4, 6); (5, 1), (5, 2), (5, 3), (5, 5), (5, 6); (6, 1), (6, 2), (6, 3), (6, 5), (6, 6)

Out of these, three ordered pairs favour the event B ∩ C ie the event that we obtain an even number on the red die and a six on the blue die. These are (2,6), (4,6), (6,6). Hence, the probability of B ∩ C is 3/36 i.e. we obtain P(B ∩ C) = 3/36

Thirdly, let us consider the tossing of the con and the two dice together: Since the coin and the two dice are being tossed together, hence, the sample space consists of 2 Ă— 6 Ă— 6 = 72 possible outcomes i.e. 72 ordered triplets. These are (H,1,1), (H,1,2), â€Ś, (T,6,6)

Out of these 72 possible outcomes, only three favour the event ‘head and even number and six’ (2,6), (4,6), (6,6). Hence, the probability of the joint event head and even number and six i.e. P(A ∩ B ∩C) is given by 3/72 i.e. P(A ∩ B ∩C) = 3/72

The equations to be verified are: P(A ∩ B) = P(A) P(B) P(A ∩ C) = P(A) P(C) P(B ∩ C) = P(B) P(C) And P(A ∩ B ∩C) = P(A) P(B) P(C)

1. Are A, B and C pairwise independent? i) P(A ∩ B) = 3/12 = 1/4 P(A) P(B) = 1/2 × 3/6 = 1/4 Hence, P(A ∩ B) = P(A) P(B) So, the events A and B are independent. ii) P(A ∩ C) = 1/12 = 1/4 P(A) P(C) = 1/2 × 1/6 = 1/12 Hence, P(A ∩ C) = P(A) P(C) So, the events A and C are independent.

iii) P(B ∩ C) = 3/36 P(B) P(C) = 3/6 × 1/6 = 3/36 Hence, P(B ∩ C) = P(B) P(C) So, the events B and C are independent.

2) Are A, B and C mutually independent? We have P(A ∩ B ∩C) = 3/72 And P(A) P(B) P(C) = 1/2 × 3/6 × 1/6 =3/72 Hence, P(A ∩ B ∩C) = P(A) P(B) P(C) and we conclude that the events A, B and C are mutually independent.

As, both the conditions of independence are fulfilled, therefore, we conclude that the events A, B and C are statistically independent.

In general, the k events A1, A2, â€Ś, Ak are defined to be mutually independent if and only if the probability of the intersection of any 2, 3, â€Ś, or k of them equals the product of their respective probabilities.

Let us now go back to the example pertaining to live births and stillbirths that we considered in the last lecture, and try to determine whether or not sex of the baby and nature of birth are independent.

EXAMPLE Table-1 below shows the numbers of births in England and Wales in 1956 classified by (a) sex and (b) whether liveborn or stillborn.

Table-1 Number of births in England and Wales in 1956 by sex and whether live- or still born. (Source Annual Statistical Review)

Male Female Total

Liveborn

Stillborn

Total

359,881 (A) 340,454 (B) 700,335

8,609 (B) 7,796 (D) 16,405

368,490 348,250 716,740

There are four possible events in this double classification: •Male livebirth, •Male stillbirth, •Female livebirth, and •Female stillbirth.

The corresponding relative frequencies are given in Table-2.

Table-2 Proportion of births in England and Wales in 1956 by sex and whether live- or stillborn. (Source Annual Statistical Review)

Liveborn

Stillborn

Total

Male

.5021

.0120

.5141

Female

.4750

.0109

.4859

.9771

.0229

1.0000

Total

As discussed in the last lecture, the total number of births is large enough for these relative frequencies to be treated for all practical purposes as PROBABILITIES.

The compound events ‘Male birth’ and ‘Stillbirth’ may be represented by the letters M and S.

If M represents a male birth and S a stillbirth, we find that

n ( M and S) 8609 = = 0.0234 n ( M) 368490

This figure is the proportion –– and, since the sample size is large, it can be regarded as the probability –– of males who are still born –– in other words, the CONDITIONAL probability of a stillbirth given that it is a male birth. In other words, the probability of stillbirths in males.

The corresponding proportion of stillbirths among females is

7796 = 0.0224. 348258

These figures should be contrasted with the OVERALL, or UNCONDITIONAL, proportion of stillbirths, which is

16405 = 0.0229. 716740

We observe that the conditional probability of stillbirths among boys is slightly HIGHER than the overall proportion. Where as the conditional proportion of stillbirths among girls is slightly LOWER than the overall proportion.

It can be concluded that sex and stillbirth are statistically DEPENDENT, that is to say, the SEX of a baby yet to be born has an effect, (although a small effect), on its chance of being stillborn.

The example that we just considered points to the concept of MARGINAL PROBABILITY. Let us have another look at the data regarding the live births and stillbirths in England and Wales:

Table-2 Proportion of births in England and Wales in 1956 by sex and whether live- or stillborn. (Source Annual Statistical Review)

Liveborn

Stillborn

Total

Male

.5021

.0120

.5141

Female

.4750

.0109

.4859

.9771

.0229

1.0000

Total

And, the figures in Table-2 indicate that the probability of male birth is 0.5141, whereas the probability of female birth is 0.4859. Also, the probability of live birth is 0.9771, where as the probability of stillbirth is 0.0229.

And since these probabilities appear in the margins of the Table, they are known as Marginal Probabilities.

According to the above table, the probability that a new born baby is a male and is live born is 0.5021 whereas the probability that a new born baby is a male and is stillborn is 0.0120. Also, as stated earlier, the probability that a new born baby is a male is 0.5141, and, CLEARLY, 0.5141 = 0.5021 + 0.0120.

Hence, it is clear that the joint probabilities occurring in any row of the table ADD UP to yield the corresponding marginal probability.

If we reflect upon this situation carefully, we will realize that this equation is totally in accordance with the Addition Theorem of Probability for mutually exclusive events.

P(male birth) = P(male live-born or male stillborn) = P(male live-born) + P(male stillborn) = 0.5021 + 0.0120 = 0.5141

Another important point to be noted is that:

Conditional Probability Joint Probability = Marginal Probability

EXAMPLE P(stillbirth/male birth) P(male birth and stillbirth) = P(male birth) = 0.0120 0.5141 = 0.0233

IN TODAY’S LECTURE, YOU LEARNT •Independent and Dependent Events •Multiplication Theorem of Probability for Independent Events •Marginal Probability

IN THE NEXT LECTURE, YOU WILL LEARN

•Bayes’ Theorem •Discrete Random Variable and its Probability Distribution

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