the probability of winning a carnival game is 15%. Elaine plays 10 times. find the probability that Elaine will win 2 times.
this is a binomial result
... win or not win
P(w) = .15 , P(n) = .85
(n + w)^10 ... you want the 3rd term
10C2 * .85^8 * .15^2
To find the probability that Elaine will win 2 times out of 10, we can use the binomial probability formula.
The binomial probability formula is given by:
P(x) = (nCx) * (p^x) * ((1-p)^(n-x))
Where:
P(x) represents the probability of x successes,
n is the number of trials (in this case, the number of times Elaine plays),
x is the number of desired successes (in this case, the number of times Elaine wins),
p is the probability of success (in this case, 15% or 0.15), and
(1-p) is the probability of failure.
Using these values, we can plug them into the formula:
P(2) = (10C2) * (0.15^2) * ((1-0.15)^(10-2))
Using combination notation,
P(2) = (10! / (2! * (10-2)!)) * (0.15^2) * (0.85^8)
P(2) = (10! / (2! * 8!)) * (0.15^2) * (0.85^8)
Calculating the factorials,
P(2) = (10 * 9) / (2 * 1) * (0.15^2) * (0.85^8)
P(2) = 45 * (0.0225) * (0.195112,...)
P(2) ≈ 0.193293
Therefore, the probability that Elaine will win 2 times out of 10 is approximately 0.193293, or about 19.33%.
To find the probability that Elaine will win 2 times out of 10 plays, we can use the binomial probability formula.
The binomial probability formula calculates the probability of getting exactly a certain number of successes in a fixed number of independent Bernoulli trials (trials with only two possible outcomes – success or failure).
In this case, the probability of winning a carnival game is 15%, which means the probability of success (p) is 0.15, and the probability of failure (q) is 1 - p = 1 - 0.15 = 0.85.
Now, we can substitute these values into the binomial probability formula:
P(X = k) = (nCk) * p^k * q^(n-k)
where:
- P(X = k) is the probability of getting exactly k successes
- n is the number of trials
- k is the number of successes
- p is the probability of success in one trial
- q is the probability of failure in one trial (= 1 - p)
In this case, n = 10 (the number of times Elaine plays the game) and k = 2 (the number of times Elaine will win).
P(X = 2) = (10C2) * (0.15)^2 * (0.85)^(10-2)
Now let's calculate this probability:
(10C2) = 10! / (2! * (10-2)!) = 45
P(X = 2) = 45 * (0.15)^2 * (0.85)^8
Calculating (0.15)^2 = 0.0225 and (0.85)^8 ≈ 0.196
P(X = 2) = 45 * 0.0225 * 0.196 ≈ 0.197 or 19.7%
Therefore, the probability that Elaine will win 2 times out of 10 plays is approximately 0.197 or 19.7%.