The probability of winning an arcade game is 0.568. If you play the arcade game 22 times, what is the probability of winning more than 15 times?

Fill in the blanks with the correct symbol [tex]$(\ \textless \ , \leq, \ \textgreater \ ,$[/tex] or [tex]$\geq$[/tex]) to represent the correct probability.

To solve this problem, we need to calculate the probability of winning more than 15 times when playing an arcade game 22 times, where the probability of winning each game is 0.568. This is a binomial distribution problem.

1. Understand the Binomial Distribution:
- The binomial distribution is used when there are fixed number of independent trials (in this case, 22 plays).
- Each trial has two possible outcomes: "success" (winning) with probability p, or "failure" (not winning) with probability 1-p.
- The probability of winning each game is given as 0.568.

2. Define the Random Variable:
- Let X be the random variable representing the number of wins in 22 games.

3. Calculate P(X > 15):
- We want to find the probability of winning more than 15 times, P(X > 15), which is the complement of winning 15 times or fewer, P(X ≤ 15).

4. Use Complement Rule:
- The complement rule states that P(X > 15) = 1 - P(X ≤ 15).
- So, we calculate 1 minus the cumulative probability of winning 15 times or fewer.

5. The Result:
- The calculated probability of winning more than 15 times, based on the data, is approximately 0.0963.

This tells us that there is roughly a 9.63% chance of winning more than 15 times when you play the arcade game 22 times.