Question

A fair coin is tossed 4 times. If $X$ is a random variable which indicates number of heads, then $P[X \lt 3] =$

• A. $\frac{10}{16}$
• B. $\frac{1}{16}$
• C. $\frac{12}{16}$
• D. $\frac{11}{16}$

Hint:

For 4 coin tosses, the total number of outcomes in the sample space is $2^4 = 16$. The counts of outcomes for $X = 0, 1, 2, 3, 4$ follow the symmetric row of Pascal's Triangle: 1, 4, 6, 4, 1. Adding the counts for $X=0, 1, 2$ gives $1 + 4 + 6 = 11$, directly yielding $\frac{11}{16}$ without using formulas!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
A fair coin is tossed 4 times; find P(X<3) where X is the number of heads.

Step 2: Key Formula or Approach:
Use binomial distribution with n=4, p=1/2; apply complement rule P(X<3) = 1 – P(X=3) – P(X=4).

Step 3: Detailed Explanation:
P(X=3)=C(4,3)(1/2)⁴=4/16; P(X=4)=C(4,4)(1/2)⁴=1/16. Sum=5/16. Complement = 1 – 5/16 = 11/16.

Step 4: Final Answer:
The probability is 11/16, matching option (D).