It's a simple question I am just so confused with this discrete probability distribution. The question is:
In the New Jersey Pick 3 game, you pay $.50 to select a sequence of three digits, such as 911. If you play this game once every day, find the probability of winning exactly once in 365 days.
To find the probability of winning exactly once in 365 days in the New Jersey Pick 3 game, we need to understand the concept of a discrete probability distribution.
A discrete probability distribution represents the probability of each possible outcome in a finite set of values. In this case, the possible outcomes are the number of times you win in 365 days, which can vary from 0 to 365.
To calculate the probability of winning exactly once in 365 days, we need to determine the total number of outcomes and the number of successful outcomes.
Total number of outcomes:
In the New Jersey Pick 3 game, you play once every day for 365 days. Each day, there are 1,000 possible three-digit numbers to choose from (000 to 999). Therefore, the total number of outcomes is 1,000^365.
Number of successful outcomes:
To win exactly once, there are two possibilities:
1. You win on a specific day and don't win on the remaining 364 days.
2. You don't win on a specific day but win on one of the remaining 364 days.
To calculate the number of successful outcomes, we multiply the number of ways to choose a specific day (365) by the number of ways to choose a specific three-digit number (1) for that day. Then we multiply it by the number of ways to not win on the remaining 364 days, which is (999/1,000)^364. Finally, we multiply this by the number of ways to win on any of the remaining 364 days, which is (1/1,000)^1 multiplied by the number of ways to choose one of those 364 days (364). Therefore, the number of successful outcomes is:
Number of successful outcomes = 365 * 1 * ((999/1,000)^364) * ((1/1,000)^1) * 364
Now, we can calculate the probability of winning exactly once by dividing the number of successful outcomes by the total number of outcomes:
Probability of winning exactly once = Number of successful outcomes / Total number of outcomes
I'll leave the computations to you. Let me know if you need any further assistance!