Question

PROBLEM 7.2 (pg 105, #20) One couple and four individuals have been invited to a dinner party where dinner will be served promptly at 8pm. The chance any individual arrives late to the party is 10%. The chance the couple arrives late is 25% (the couple will arrive together). Let X = the number of guests arriving late. Note: suppose I wanted to find the probability that 3 people arrive late to the party. This can happen in two ways: • (a) any combination of 3 individuals arrive late (all others on time) or • (b) any 1 individual and the couple arrive late (all others on time). We consider in the following table all ways that (a) any combination of 3 individuals are late, the couple is on-time, and the remaining individual is on-time: Couple Individual Individual Individual Individual #1 #2 Probability #3 #4 On Time Late Late Late On Time 0.75(0.1)(0.9) (0.75) (0.1) (0.1) (0.1) (0.9) On Time Late Late On Time Late 0.75(0.1) '(0.9) (0.75) (0.1) (0.1) (0.9) (0.1) On Time Late On Time Late 0.75(0.1) (0.9) (0.25) (0.1) (0.1) On Time On Time Late Late Late 0.75(0.1) (0.9) (0.75) (0.9) (0.1) This part of the table can consist of any arrangement of Late Late Late, OnTime. Must be 4! (0.75)(0.1) (0.9) OnTime 3!1! There are 3!1! such arrangements. Like the Mississippi problem in powerpoint #1. So the probability that 3 people arrive late is P( X =3) = 4!(0.75)(0.1) (0.9)' + 4!(0.25)(0.1)'(0.9) Late 10.9) (0.1) (0.1) (0.1) 3!1! Find: a. Find the chance that everyone is on time or PCX = 0) b. Find the chance that exactly four people are late or PCX = 4). Hint: two cases: 1) couple is on-time and all individuals are late. 2) couple is late and any two individuals are late.

please explain this process for my understanding.

Answer 1

chance that every one is on time le P(x=0)= P(couple is are - P( couple is on on on time and all individuals tine) time). P{all individueals are on time) = 0.75 0.9 - 0.675 6 chance that the exactly four people are lote = P(x=4) - Pl couple is on ting and all individuals are late or couple is late and any two individuals are late) - P(couple is on time ) . plall individuals are late) + P( couple is late). Plany two individuals are late) - (0.75) (0.1) + (0.25) 4 2 (0.1)? Co.g² - 0.075 +0.01215 = 0.08715