Step 1: Concept Overview:
The problem involves transforming a discrete random variable, X, into a new variable, Y. The goal is to find the probability of an event defined in terms of Y. This is achieved by first expressing the event in terms of X and then using the probability mass function (PMF) of X to compute the probability.
Step 2: Solution Strategy:
1. Express the inequality involving Y as an equivalent inequality involving X: \( Y \le 6 \implies X^2+2 \le 6 \).
2. Determine the integer values of X that satisfy the inequality.
3. Compute the probabilities for these X values using the Binomial PMF: \( P(X=k) = \binom{n}{k}p^k(1-p)^{n-k} \).
4. Sum the calculated probabilities.
Step 3: Detailed Solution:
1. Event Translation: \[ Y \le 6 \] \[ X^2 + 2 \le 6 \] \[ X^2 \le 4 \] Since X represents the number of successes, it is a non-negative integer. The integer solutions to \(X^2 \le 4\) are \(X = 0, 1, 2\).
2. The problem is now reduced to calculating \( P(X \le 2) = P(X=0) + P(X=1) + P(X=2) \).
3. Apply the Binomial PMF with \(n=8\) and \(p=1/2\). Notice that \(p^k(1-p)^{n-k}\) simplifies to \( (1/2)^k (1/2)^{8-k} = (1/2)^8 = 1/256 \).
- \( P(X=0) = \binom{8}{0} \left(\frac{1}{2}\right)^8 = 1 \times \frac{1}{256} = \frac{1}{256} \)
- \( P(X=1) = \binom{8}{1} \left(\frac{1}{2}\right)^8 = 8 \times \frac{1}{256} = \frac{8}{256} \)
- \( P(X=2) = \binom{8}{2} \left(\frac{1}{2}\right)^8 = \frac{8 \times 7}{2 \times 1} \times \frac{1}{256} = 28 \times \frac{1}{256} = \frac{28}{256} \)
4. Summation of Probabilities: \[ P(Y \le 6) = P(X \le 2) = \frac{1}{256} + \frac{8}{256} + \frac{28}{256} = \frac{37}{256} \]
5. Decimal Conversion: \[ \frac{37}{256} \approx 0.1445 \]
The calculated probability of 0.1445 is not directly available among the answer options. 0.165 is the closest value. It is possible that there is an error in the question or the answer choices. The closest answer would be the most appropriate selection in an examination setting.
Step 4: Conclusion:
The calculated probability is approximately \(37/256 \approx 0.1445\). The nearest provided option is 0.165.