probability question

Question

The probability of buying a movie ticket with a popcorn coupon is 0.629 and without a popcorn coupon is 0.371. If you buy 29 movie tickets, we want to know the probability that more than 16 of the tickets have popcorn coupons.

Answer 1

This is easy to solve if you have the combination function and summation function in your calculator. I'll show you both with and without calculator:

Let:

X = 0.629

Y = 0.371

n = 29

X + Y = 1

If we expand (X + Y)²⁹, we'll have:

X²⁹ + 29X²⁸Y + 406X²⁷y² + ... + 406X²Y²⁷ + 29XY²⁸ + Y²⁹

There are 30 terms in this expression. Since we are only looking for more than 16 of the tickets have popcorns, we only have to add terms with X¹⁷, X¹⁸, ... ,X²⁹.

We can use also combination for the coefficient of each term:

₂₉C₀•X²⁹Y⁰ = X²⁹

₂₉C₁•X²⁸Y¹ = 29X²⁸Y

₂₉C₂•X²⁷Y² = 406X²⁷Y²

.

.

.

₂₉C₁₂•X¹⁷Y¹² = 51895935X¹⁷Y¹²

Plugin the value of X and Y then add it all up. The sum will be the final answer.

Or you can do it this way with your calculator using sum and combination functions:

∑²⁹_n=17 (₂₉C_29-n)•(0.629)ⁿ (0.371)^29-n ≈ 0.751