a)One game in a state lottery requires you to pick 1 heart, 1 club, 1 diamond, and 1 spade, in that order, from the 13 cards in each suit. What is the probability of getting all four picks correct and winning $5000

b)If three of the four selections in part (a) are correct, the player will win $200. What is the probability of this outcome?

1/4

a) To find the probability of getting all four picks correct and winning $5000, we need to calculate the probability of each individual pick and multiply them together.

There are 13 cards in each suit, so the probability of picking the correct heart is 1/13. After picking the correct heart, there are 12 cards left in the suit, so the probability of picking the correct club is 1/12. Similarly, after picking the correct club, there are 11 cards left in the suit, so the probability of picking the correct diamond is 1/11. Finally, after picking the correct diamond, there are 10 cards left in the suit, so the probability of picking the correct spade is 1/10.

To calculate the overall probability, we multiply these individual probabilities together:

Probability = (1/13) * (1/12) * (1/11) * (1/10) = 1/17160

Therefore, the probability of getting all four picks correct and winning $5000 is 1/17160.

b) Similarly, to find the probability of getting three out of four picks correct and winning $200, we need to consider the different possible combinations of getting three picks correct.

There are four picks in total, so there are four different scenarios where three picks are correct (heart, club, diamond or heart, club, spade or heart, diamond, spade or club, diamond, spade). In each scenario, there are still nine cards left in the suit from which the incorrect pick is made.

So, for each scenario, the probability of getting three picks correct is the product of the individual probabilities of each correct pick and the probability of the incorrect pick:

Probability of each scenario = (1/13) * (1/12) * (1/11) * (9/10)

Since there are four scenarios, we multiply this probability by four:

Probability = 4 * (1/13) * (1/12) * (1/11) * (9/10) = 9/1430

Therefore, the probability of getting three out of four picks correct and winning $200 is 9/1430.