Question

Suppose you have 2 nickels, 3 dimes, and 1 quarter in a coin purse. You choose two coins from the purse at random. What is the probability you draw out at least 20 cents?

A three of a kind in a five-card poker hand is obtained by having exactly three of the same type of card (3 Js or 3 4s). What is the probability that a randomly selected five-card hand is a three of a kind?

Answer 1

1. Given 2 nickels, 3 dimes, and 1 quarter in a coin purse
Draw 2 coins from the purse at random
The number of ways draw 2 coins from the purse (3 + 2 + 1 = 6) is 6C₂ = 15
The probability that you draw out at least 20 cents is
P(1 nickels and 1 quarter) + P(2 dimes) + P(1 dimes 1 quarter)
= (2*3)/ 15 + (3C₂)/ 15 + (3*1)/ 15
= (6+3+3)/15
= 12/15
= 0.8

2. a) Since there are 52 cards in the deck, then there are 52C5 = 2,598,960 possible
combinations of five-card hands possible

The number of possible ways of exactly 3 of the same type of card (3 Js or 3 4s) is
P(3 Js and 2 remaining is any other cards) + P(3 4s and remaining is any other cards)
= 4C3 * 48C2 + 4C3 * 48C₂
= 9024

Required probability = 9024 / 2598960 = 0.00347

b) The probability that a randomly selected five-card hand is a three of a kind is

The number of possible ways of exactly 3 of the same kind is
P(3 of one kind and 2 remaining is any other cards)
= 13 * 4C₃ * 48C₂
= 58656

Required probability = 58656 / 2598960 = 0.02257