A weighted coin is biased so that a head is twice as likely

A weighted coin is biased so that a head is twice as likely to occur as a tail. If the coin is tosses 4 times, what is the probability of getting at least one head?

Solution

Given: a head is twice as likely to occur as a tail

Probability of getting a head = 2 * Probability of getting a tail

P(Head) = 2 * P(Tail)

Since P(Head) + P(Tail) = 1

2P(Tail) + P(Tail) = 1

P(Tail) = 1/3

P(Head) = 2/3

probability of getting no heads for 4 tosses = 1/3 * 1/3 * 1/3 * 1/3 = 1/81

probability of getting at least one head = 1 - probability of getting no heads

probability of getting at least one head = 1 - 1/81 = 80/81