A fair coin is tossed until a head or five tails occur. The probability distribution of the number of tosses is,
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → X 1 2 3 4 5 P(X) 1/2 1/4 1/8 1/16 1/16
Let X be the number of tosses. The tossing stops at the first head, or after the fifth toss if the first five are all tails, so X takes the values 1, 2, 3, 4, 5.
X = k (for k = 1 to 4) means k − 1 tails followed by a head:
\( \displaystyle P(X=1)=\frac{1}{2} \)
\( \displaystyle P(X=2)=\frac{1}{4} \)
\( \displaystyle P(X=3)=\frac{1}{8} \)
\( \displaystyle P(X=4)=\frac{1}{16} \)
X = 5 whenever the first four tosses are tails, whatever the fifth toss shows:
\( \displaystyle P(X=5)=\left(\frac{1}{2}\right)^{4} \)
\( \displaystyle =\frac{1}{16} \)
As a check, \( \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{16}=1 \).