Binomial Distribution

Question

1) The ratio of the probability of 3 successes in 5 independent trials to the probability of 2 successes in 5 independent trails is 1/4 , what is the probability of 4 success in 6 independent trials. ?

2) out of 800 families with 5 children each how many would you expect to have.?

(i) 3 boys & 2 girls

(ii) 5 girls

(iii) 5 boys

Answer 1

Probability of 3 successes in 5 trials (per binomial probability formula):

C(5,3) p^3 (1-p)^2 = 10 * p^3 (1-p)^2

Probability of 2 successes in 5 trials (per binomial probability formula):

C(5,2) p^2 (1-p)^3 = 10 * p^2 (1-p) ^2

ratio of probabilities:

10 * p^3 (1-p)^2 p 1

---------------------- = ------ = ----

10 * p^2 (1-p)^3 1-p 4

4p = 1 - p

3p = 1

p = 1/3

Probability of 4 successes in 6 trials (per binomial probability formula), using p = 1/3

C(6,4) (1/3)^4 (2/3)^2 = 15 * 1/81 * 4/9 = 60/729 = 20/243:

n = 5, p = 0.5 (even probability of having a boy or girl)

probability of having 3 boys and 2 girls

C(5,3)(0.5)^5 = 0.3125

number of families = 800 * 0.3125 = 250

Probability of having 5 girls

C(5,5)(0.5)^5 = 0.03125

number of families = 800 * 0.03125 = 25

Probability of having 5 boys

C(5,5)(0.5)^5 = 0.03125

number of families = 800 * 0.03125 = 25