Unit 3, Topic 11, Exercise 11-12 Family Births Suppose a couple has four childre

Question

Unit 3, Topic 11, Exercise 11-12 Family Births Suppose a couple has four children. Assume that each child is equally likely to be a boy or a girl, regardless of the outcome of previous births. The following table shows the results of a simulation to determine the gender breakdown in 20 familles. Number of girls 6 Probsblity 205-0303025 20-0 a. Based on the simulation results, which is more likely, a 2:2 split of boys and girls or a 3:1 split in either direction? split is more likely with a probability of b. Write out the sample space and calculate the exact probabilities for the various possible gender breakdowns. Enter the exact answer as a decimal. 4 Number of girls Probability c. Reanswer part a. using exact probabilities. split is more likely with a probability of Click if you would like to Show Work for this question: Open Show Work

Explanation / Answer

Let X = Number of girl children in 4-children families. Then, given that each child is equally likely to be a boy or a girl, X ~ B(4, 1/2) …………………………………..(A)

Back-up Theory

If X ~ B(n, p). i.e., X has Binomial Distribution with parameters n and p, where n = number of trials and p = probability of one success, then

probability mass function (pmf) of X is given by

p(x) = P(X = x) = (nCx)(px)(1 - p)n – x, x = 0, 1, 2, ……. , n …………………..(1)

[The above probability can also be directly obtained using Excel Function of Binomial Distribution: BINOMDIST(Number_s:Trials:Probability_s:Cumulative), what is within brackets is (x:n:p:True)] ………………………………………………………………….(1a)

Part (a)

For 2 : 2 split of boys and girls, probability = P(2 girls) = 0.3

For 3 : 1 split of boys and girls in either direction, probability = P(3 girls) + P(1 girl)

= 0.25 + 0.3 = 0.55

Since 0.55 > 0.3, 3 : 1 split is more likely with probability 0.55 ANSWER

Part (b)

Sample space: {0, 1, 2, 3, 4}

Theoretical probabilities [vide (A) and (1)]

x

0

1

2

3

4

p(x)

1/16

¼

3/8

¼

1/16

ANSWER

Part (c)

Based on theoretical probabilities,

For 2 : 2 split of boys and girls, probability = P(2 girls) = 3/8

For 3 : 1 split of boys and girls in either direction, probability = P(3 girls) + P(1 girl)

= 0.25 + 0.25 = 0.5

Since 0.5 > 3/8, 3 : 1 split is more likely with probability 0.5 ANSWER

x

0

1

2

3

4

p(x)

1/16

¼

3/8

¼

1/16

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