A coin is tossed 3 times. If X denotes the number of heads, then Match List-I with List-II.
| List-I | List-II | ||
| A. | P(X = 0) | I. | \(\displaystyle\frac{1}{8}\) |
| B. | P(X = 1) | II. | \(\displaystyle\frac{7}{8}\) |
| C. | \(P(X \le 2)\) | III. | \(\displaystyle\frac{4}{8}\) |
| D. | \(P(X \ge 2)\) | IV. | \(\displaystyle\frac{3}{8}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → A-I, B-IV, C-II, D-III
X follows the binomial distribution with n = 3 and \(p = \frac{1}{2}\):
\( \displaystyle P(X = r) = {}^{3}C_{r}\left(\frac{1}{2}\right)^3 = \frac{{}^{3}C_{r}}{8} \)
\( \displaystyle P(0) = \frac{1}{8},\ P(1) = \frac{3}{8},\ P(2) = \frac{3}{8},\ P(3) = \frac{1}{8} \)
(A) P(X = 0)
This matches (I) \(\frac{1}{8}\): only TTT has no head.
(B) P(X = 1)
This matches (IV) \(\frac{3}{8}\): HTT, THT, TTH.
(C) \(P(X \le 2)\)
This matches (II) \(\frac{7}{8}\).
\( \displaystyle P(X \le 2) = 1 - P(X = 3) = 1 - \frac{1}{8} = \frac{7}{8} \)
(D) \(P(X \ge 2)\)
This matches (III) \(\frac{4}{8}\).
\( \displaystyle P(X \ge 2) = P(2) + P(3) = \frac{3}{8} + \frac{1}{8} = \frac{4}{8} \)