Question

The probability of winning on an arcade game is 0.659. if you play the arcade game 30 times. What is the probability of winning exactly 21 times?

1. The probability of winning exactly 21 times is 0.14

### What is Binomial Probability?

Binomial probability refers to the probability of exactly ‘x’ successes on ‘n’ repeated trials in an experiment which has two possible outcomes (commonly called a binomial experiment).
Probability ‘P’ = ⁿCₓ (probability of 1st)ˣ x (1 – probability of 1st)ⁿ⁻ˣ
For example:
What is the probability of getting 6 heads, when you toss a coin 10 times?
In a coin-toss experiment, there are two outcomes: heads and tails. Assuming the coin is fair , the probability of getting a head is 1/2 or 0.5 .
The number of repeated trials: n=10
The number of success trials: x = 6
The probability of success on individual trial: p = 0.5
Use the formula for binomial probability.
¹⁰C₆ (0.5)⁶ x (1 – 0.5)¹⁰⁻⁶
Simplify.
≈0.205
Here, we have given that:
Probability of winning on an arcade game is 0.659
so, Probability of loosing on an arcade game is 1-0.659 = 0.341
Number of times game played = 30
Probability of winning exactly 21 times = ?
now, n = 30
x = 21
probability of 1st or probability of winning = 0.659
1 – probability of 1st or probability of loosing = 0.341
using, binomial probability formula
Probability of winning exactly 21 times = ³⁰C₂₁ (0.659)²¹ x (0.341)⁷
on solving,
Probability of winning exactly 21 times = 0.14
Hence,
The probability of winning exactly 21 times is 0.14