Mr. Patterson's coins:
Without replacement:
P(quarter or penny): Since there are 6 quarters and 8 pennies, and Mr. Patterson isn't putting any coins back in, there are 14 favorable outcomes (quarters or pennies) out of 21 total possible outcomes (all the coins). Therefore, the probability is 14/21.
P(1st is a dime and 2nd is a penny): There are 3 dimes and 8 pennies, so 3 * 8 = 24 favorable outcomes (dime then penny). There are 21 total possible outcomes since Mr. Patterson isn't replacing any coins. Therefore, the probability is 24/21 which reduces to 8/7.
With replacement:
P(1st is a nickel and 2nd is a quarter): Since Mr. Patterson is putting the coins back in each time, each selection is independent of the other. So, the probability of getting a nickel is 3 out of 21, and the probability of getting a quarter is 6 out of 21. Therefore, the probability of getting a nickel then a quarter is 3/21 * 6/21, which reduces to 2/7.
P(dime or a penny): There are 3 dimes and 8 pennies, so 11 favorable outcomes (dimes or pennies) out of 21 total possible outcomes. Therefore, the probability is 11/21.
Without replacement, selecting 3 coins:
P(1st is a penny, 2nd is a quarter, and 3rd is a penny): There are 8 ways to pick a penny first, 6 ways to pick a quarter second (given Mr. Patterson isn't putting the first coin back in), and then 7 ways to pick another penny third (given Mr. Patterson isn't putting any of the first two coins back in). So, there are 8 * 6 * 7 = 336 favorable outcomes. There are a total of 212019 = 7980 possible outcomes (picking any 3 coins without replacement). Therefore, the probability is 336/7980 which reduces to 2/45.
Standard deck of cards:
P(ace or a black card): There are 4 aces, 26 black cards (all the spades and clubs), and 52 cards total. So, there are 4 + 26 = 30 favorable outcomes. Therefore, the probability is 30/52 which reduces to 15/26.
P(6 or a red card): There is 1 red 6, 13 red cards total, and 52 cards total. So, there are 14 favorable outcomes. Therefore, the probability is 14/52 which reduces to 7/26.
P(red 5): There is 1 red 5 and 52 cards total. Therefore, the probability is 1/52.
P(A Joker): There is 1 joker and 52 cards total. Therefore, the probability is 1/52.
