Question

5. Prince Charming has to go around town asking if a glass slipper fits until he finds 10 women whose foot fits properly in a slipper (they will compete on a "Bachelor Type" show) The probability of a glass slipper fitting a randomly selected woman is 0.12, and the probabilities are independent. Let X be the number of women he has to visit until he finds a woman who fits the slipper.

a) What is the probability he has to check 75 women?

b) Given that he has to check between 75 and 77 women, what is the probability he has to check exactly 76 women?

Answer 1

X is the number of women he has to visit until he finds a woman who fits the slipper. Therefore X here has the following distribution:

a) Probability that he has to check 75 women is computed here as:

= Probability that for first 74 women, glass slipper does not fit * Probability that the slipper fits for 75^th woman
= (1 - p)⁷⁴*p
= (1 - 0.12)⁷⁴*0.12
= 0.00000935

Therefore 0.00000935 is the required probability here.

b) Given that he has to check between 75 and 77 women, probability that he has to check exactly 76 women is computed here as:
= P(X = 76) / [P(X = 75) + P(X = 76) + P(X = 77)]
= (1-p)⁷⁵p / p(1-p)⁷⁴(1 + (1-p) + (1-p)²)
= (1-p) / (2 - p + 1 + p² - 2p)
= (1-p) / (3 - 3p + p²)
= (1 - 0.12) / (3 - 3*0.12 + 0.12²) = 0.3315

Therefore 0.3315 is the required probability here.