Question

What is the probability of getting fewer than 2 successes in 6 trials if the success probability is 0.5?

• A. 0.016
• B. 0.109
• C. 0.234
• D. 0.344

Answer 1

Answer: (B)

0.109

Answer 2

The problem describes a binomial distribution scenario.

1. Identify the parameters:

• Number of trials ($n$) = 6
• Probability of success ($p$) = 0.5
• Probability of failure ($q = 1 - p$) = 1 - 0.5 = 0.5

2. Determine the required events: We need to find the probability of getting fewer than 2 successes. This means the number of successes ($X$) can be 0 or 1. So, we need to calculate $P(X < 2) = P(X=0) + P(X=1)$.

3. Apply the binomial probability formula: The probability mass function for a binomial distribution is given by: $P(X=k) = \binom{n}{k} p^k q^{n-k}$

• For $P(X=0)$: $P(X=0) = \binom{6}{0} (0.5)^0 (0.5)^{6-0}$ $P(X=0) = 1 \times 1 \times (0.5)^6$ $P(X=0) = (0.5)^6 = \frac{1}{64}$ $P(X=0) = 0.015625$

• For $P(X=1)$: $P(X=1) = \binom{6}{1} (0.5)^1 (0.5)^{6-1}$ $P(X=1) = 6 \times (0.5)^1 \times (0.5)^5$ $P(X=1) = 6 \times (0.5)^6$ $P(X=1) = 6 \times \frac{1}{64} = \frac{6}{64} = \frac{3}{32}$ $P(X=1) = 0.09375$

4. Sum the probabilities: $P(X < 2) = P(X=0) + P(X=1)$ $P(X < 2) = 0.015625 + 0.09375$ $P(X < 2) = 0.109375$

5. Compare with the given options: The calculated probability $0.109375$ is approximately $0.109$.