if a pitcher is just as likely to throw a ball as a strike, find the probability that t11 of his first pitches are balls.
To find the probability that exactly t11 of the pitcher's first pitches are balls, we need to use the binomial probability formula.
The probability of a pitch being a ball is 0.5, as the pitcher is equally likely to throw a ball as a strike.
The binomial probability formula is given by:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
Where:
P(X = k) is the probability of getting exactly k successes (in this case, exactly t11 balls)
n is the total number of trials (in this case, the number of first pitches)
k is the number of successes (in this case, the number of balls)
p is the probability of success (in this case, the probability of getting a ball)
Let's calculate the probability.
To find the probability that exactly t11 out of the first pitches are balls, we need to use the concept of binomial probability.
The binomial probability formula is given by:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of getting exactly k successes
- C(n, k) is the number of combinations of n items taken k at a time (also known as "n choose k" or binomial coefficient)
- p is the probability of success for each trial (in this case, the probability of throwing a ball)
- n is the total number of trials (in this case, the total number of pitches)
In our case, since the pitcher is equally likely to throw a ball or a strike, the probability of throwing a ball (p) is 0.5, and the probability of throwing a strike is also 0.5.
Let's calculate the probability of exactly t11 out of the first pitches being balls.