Answer:To find the probability that exactly 2 out of 8 friends win a free hamburger, we can use the binomial probability formula. The binomial probability formula is:
P(x) = (nCx) * p^x * (1-p)^(n-x)
Where:
- P(x) represents the probability of exactly x successes,
- nCx represents the number of combinations of n items taken x at a time,
- p represents the probability of success for each trial, and
- n represents the total number of trials.
In this case, we have:
- n = 8 (since there are 8 friends),
- p = 0.19 (since there is a 19% chance of winning a free hamburger), and
- x = 2 (since we want to find the probability of exactly 2 friends winning a free hamburger).
To calculate the probability, we substitute these values into the formula:
P(2) = (8C2) * 0.19^2 * (1-0.19)^(8-2)
Step-by-step explanation: Let's solve this step by step:
1. Calculate (8C2), which represents the number of combinations of 8 items taken 2 at a time:
(8C2) = 8! / (2! * (8-2)!)
= 8! / (2! * 6!)
= (8 * 7) / (2 * 1)
= 28
2. Calculate 0.19^2, which represents the probability of exactly 2 successes:
0.19^2 = 0.0361
3. Calculate (1-0.19)^(8-2), which represents the probability of exactly 6 failures:
(1-0.19)^(8-2) = 0.81^6 = 0.2824
4. Substitute these values back into the formula:
P(2) = 28 * 0.0361 * 0.2824
= 0.2884
Therefore, the probability that exactly 2 out of 8 friends win a free hamburger is approximately 0.2884, or 28.84%.