Questions: 3. a binomial distribution with 3 trials and a success probability of 0.5. what would be the probability of a success on every trial?

Solution Steps

We are given a binomial distribution with the following parameters:

• Number of trials \( n = 3 \)
• Probability of success on each trial \( p = 0.5 \)
• Probability of failure on each trial \( q = 1 - p = 0.5 \)

To find the probability of achieving success on every trial (i.e., \( x = 3 \)), we use the binomial probability formula:

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

Substituting the values into the formula:

\[ P(X = 3) = \binom{3}{3} \cdot (0.5)^3 \cdot (0.5)^{3-3} \]

Calculating each component:

• \( \binom{3}{3} = 1 \)
• \( (0.5)^3 = 0.125 \)
• \( (0.5)^0 = 1 \)

Thus, we have:

\[ P(X = 3) = 1 \cdot 0.125 \cdot 1 = 0.125 \]

The probability of success on every trial is:

\[ P(X = 3) = 0.125 \]

Final Answer