Question

A binomial experiment is conducted with n= 32, p = 0.34, and x = 10. Copmpute the probability of x successes in

then independent trials.
Round your answer to four decimal places.
P(x: = 10) is

Answer 1

The probability of getting exactly 10 successes in 32 independent trials, where the probability of success on each trial is 0.34, is approximately 0.1427 when rounded to four decimal places.

Explanation:

1. Identify the parameters of the binomial experiment. In this case, we have:

-
`\(n = 32\)`, the number of trials

-
`\(p = 0.34\)`, the probability of success on each trial

-
`\(x = 10\)`, the number of successes we're interested in

2. Use the formula for the probability mass function (PMF) of a binomial distribution. The PMF gives the probability of getting exactly
`\(x\)` successes in
`\(n\)` trials:

`\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot (1-p)^(n-x) \]`

Where
`\(\binom{n}{x}\)` is the binomial coefficient, also known as "n choose x". It represents the number of ways to choose
`\(x\)` successes out of
`\(n\)` trials.

3. Substitute the given values into the formula. In this case, we have:

`\[ P(X = 10) = \binom{32}{10} \cdot 0.34^(10) \cdot (1-0.34)^(32-10) \]`

4. Calculate the binomial coefficient. The binomial coefficient
`\(\binom{n}{x}\)` can be calculated as:

`\[ \binom{n}{x} = (n!)/(x!(n-x)!) \]`

Where
`\(n!\)` is the factorial of
`\(n\)`, which is the product of all positive integers up to
`\(n\)`.

5. Calculate the remaining parts of the formula. This involves raising
`\(p\)` to the power of
`\(x\)`, and raising
`\((1-p)\)` to the power of
`\((n-x)\)`.

6. Multiply all the calculated values together. This gives the final probability.

The result is approximately 0.1427.