A high school baseball player has a 0.193 batting average. In one game, he gets 7 at bats. What is the probability he will get at least 2 hits in the game?
To find the probability that the high school baseball player will get at least 2 hits in the game, we can use the binomial probability formula.
The binomial probability of getting exactly x successes (hits) in n independent trials (at bats) can be calculated using the following formula:
P(x) = C(n, x) * p^x * (1 - p)^(n - x)
Where:
- P(x) is the probability of getting exactly x hits,
- C(n, x) is the number of combinations of n items taken x at a time (given by n! / (x! * (n - x)!)),
- p is the probability of getting a hit on one at-bat, and
- n is the total number of at-bats.
In this case, the baseball player has a batting average of 0.193, which means he succeeds (gets a hit) in 19.3% of his at-bats. Therefore, the probability of getting a hit in one at-bat (p) is 0.193, and the total number of at-bats (n) is 7.
Now, to calculate the probability of getting at least 2 hits, we need to find the probabilities of getting 2, 3, 4, 5, 6, and 7 hits and sum them up.
P(at least 2 hits) = P(2 hits) + P(3 hits) + P(4 hits) + P(5 hits) + P(6 hits) + P(7 hits)
To calculate each probability, substitute the values into the binomial probability formula and perform the calculations. Finally, add up the individual probabilities to find the overall probability of getting at least 2 hits in the game.
To calculate the probability that the high school baseball player will get at least 2 hits in the game, we need to use the binomial probability formula.
The formula for the probability of getting exactly k successes in n trials, where the probability of success in each trial is p, is given by:
P(k) = (nCk) * p^k * (1-p)^(n-k)
where nCk is the number of combinations of n items taken k at a time, p is the probability of success, and (1-p) is the probability of failure.
In this case, the player has a batting average of 0.193, which means the probability of getting a hit in any given at bat is 0.193.
Using the formula, we can calculate the probability of getting at least 2 hits in a game with 7 at bats:
P(at least 2 hits) = 1 - P(0 hits) - P(1 hit)
Now let's calculate each term step by step:
P(0 hits) = (7C0) * (0.193^0) * (0.807^7)
= 1 * 1 * 0.0878
= 0.0878
P(1 hit) = (7C1) * (0.193^1) * (0.807^6)
= 7 * 0.193 * 0.1753
= 0.2365
P(at least 2 hits) = 1 - P(0 hits) - P(1 hit)
= 1 - 0.0878 - 0.2365
= 0.6757
Therefore, the probability that the high school baseball player will get at least 2 hits in the game is approximately 0.6757 or 67.57%.
this is a binomial problem
probability of hit (h) = .193
probability of no hit (n) = 1 - .193 = .807
at least 2 means not zero and not one
(n + h)^7 = n^7 + 7 n^6 h...
p(h ≥ 2) = .807^7 + (7 * .807^6 * .193)