Question

A biased coin is such that the probability of getting a head is thrice the probability of getting a tail. If the coin is tossed 4 times, what will be the probability of getting a head all the times?

• A. \( \frac{16}{81} \)
• B. \( \frac{27}{64} \)
• C. \( \frac{81}{128} \)
• D. \( \frac{81}{256} \)

Hint:

For biased coins, carefully express the probabilities and then apply the multiplication rule for independent events to calculate the likelihood of multiple outcomes.

Answer 1

The Correct Option is D

Let the probability of getting a tail be \( p \).
Then, the probability of getting a head is \( 3p \).
Since the total probability must sum to 1: \[ p + 3p = 1 \quad \Rightarrow \quad 4p = 1 \quad \Rightarrow \quad p = \frac{1}{4} \] Thus, the probability of getting a head is: \[ 3p = 3 \times \frac{1}{4} = \frac{3}{4} \] The probability of getting a head all the times in 4 tosses is: \[ \left( \frac{3}{4} \right)^4 = \frac{81}{256} \]