Questions: Match the binomial probability P(x<23) with the correct statement. A. P (there are fewer than 23 successes) B. P (there are at most 23 successes) C. P (there are at least 23 successes) D. P (there are more than 23 successes)

Match the binomial probability P(x<23) with the correct statement.
A. P (there are fewer than 23 successes)
B. P (there are at most 23 successes)
C. P (there are at least 23 successes)
D. P (there are more than 23 successes)

Transcript text: Match the binomial probability $\mathrm{P}(\mathrm{x}<23)$ with the correct statement. A. P (there are fewer than 23 successes) B. $P$ (there are at most 23 successes) C. P (there are at least 23 successes) D. P (there are more than 23 successes)

Solution

Solution Steps

Step 1: Understanding the Problem

We need to match the binomial probability $ P(X < 23) $ with the correct statement among the given options. The options are:

A. $ P $ (there are fewer than 23 successes)
B. $ P $ (there are at most 23 successes)
C. $ P $ (there are at least 23 successes)
D. $ P $ (there are more than 23 successes)

Step 2: Calculating the Probability

To find $ P(X < 23) $, we recognize that this is equivalent to calculating the cumulative probability of having fewer than 23 successes. Specifically, we can express this as:

\[ P(X < 23) = P(X \leq 22) \]

Using the binomial probability formula, we calculate $ P(X = 22) $:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

For our case, with $ n = 30 $, $ x = 22 $, $ p = 0.5 $, and $ q = 0.5 $, we find:

\[ P(X = 22) = 0.0055 \]

Step 3: Matching the Statement

Since $ P(X < 23) $ corresponds to the probability of having fewer than 23 successes, we can directly match it with option A:

\[ P(X < 23) = P(X \text{ has fewer than } 23 \text{ successes}) \]