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Chapter 11: Problem 22

Write and solve an original problem involving probability and permutations.

Short Answer

Expert verified
The probability of drawing out 5 cards such that there will be atleast 2 aces is given by the formula [C(4,2) * C(48,3) + C(4,3) * C(48,2) + C(4,4) * C(48,1) + C(4,5)] / C(52,5).

Step by step solution

01

Problem Formulation

Let's consider a scenario where we have a deck of 52 playing cards. We want to calculate the probability that when drawing 5 cards, we get atleast 2 aces and the rest are random cards.
02

Defining the Problem using Combinations and Permutations

The total number of ways to draw 5 cards out of 52 is given by C(52,5). This is a combinations problem because the order of the cards drawn doesn't matter. Now, getting exactly 2 aces out of 4 aces in 5 cards can be given by C(4,2) * C(48,3), which means choose 2 aces from 4 aces and the other 3 cards can be any from the 48 remaining ones. Similarly for exactly 3 aces it will be C(4,3) * C(48,2), for exactly 4 aces it will be C(4,4) * C(48,1) and for all 5 aces it will be C(4,5).
03

Calculating the Probability

We use the formula of probability which is P(E) = favorable outcomes / total outcomes. The total outcomes is C(52,5). The favorable outcomes is the sum of the scenarios where we have exactly 2, 3, 4 or 5 aces. So the probability will be [C(4,2) * C(48,3) + C(4,3) * C(48,2) + C(4,4) * C(48,1) + C(4,5)] / C(52,5).

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