5. Binomial Distribution & Discrete Random Variables
Hypergeometric Distribution
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Find the probability of drawing a hand of 5 cards from a standard deck that contains exactly 2 hearts.
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Verified step by step guidance1
Step 1: Understand the problem. You are tasked with finding the probability of drawing a hand of 5 cards from a standard deck of 52 cards that contains exactly 2 hearts. A standard deck has 13 hearts and 39 non-hearts.
Step 2: Use the combination formula to calculate the number of ways to choose 2 hearts from the 13 available hearts. The combination formula is given by: , where n is the total number of items, and r is the number of items to choose.
Step 3: Similarly, calculate the number of ways to choose 3 non-hearts from the 39 available non-hearts using the same combination formula.
Step 4: Multiply the results from Step 2 and Step 3 to find the total number of favorable outcomes. This is because the events of choosing hearts and non-hearts are independent.
Step 5: Divide the total number of favorable outcomes by the total number of ways to choose 5 cards from the deck (calculated using the combination formula with n=52 and r=5). This gives the probability of drawing a hand with exactly 2 hearts.
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