If a coin is balanced so that p(Heads)= p(Tails)= 1/2,then which of the following outcomes is least likely to occur?
A)9 Heads in 12 tosses (75% Heads)
B)12 Heads in 16 tosses (75% Heads)
C)30 Heads in 40 tosses (75% Heads)
D)75 Heads in 100 tosses (75% Heads)

Question

If a coin is balanced so that p(Heads)= p(Tails)= 1/2,then which of the following outcomes is least likely to occur?​
A)​9 Heads in 12 tosses (75% Heads)
B)​12 Heads in 16 tosses (75% Heads)
C)​30 Heads in 40 tosses (75% Heads)
D)​75 Heads in 100 tosses (75% Heads)

Quizplus · Accepted Answer

The binomial distribution is a probability distribution that describes the number of successes in a sequence of independent experiments, each of which has a constant probability of success. The probability of getting exactly k heads in n tosses of a fair coin is given by the binomial distribution:$$P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}$$where p is the probability of getting heads on each toss.In this case, p = 1/2, so the probability of getting exactly k heads in n tosses is:$$P(X = k) = \binom{n}{k}\left(\frac{1}{2}\right)^k\left(\frac{1}{2}\right)^{n-k} = \binom{n}{k}\left(\frac{1}{2}\right)^n$$The probability of getting 9 heads in 12 tosses is:$$P(X = 9) = \binom{12}{9}\left(\frac{1}{2}\right)^{12} = 0.2266$$The probability of getting 12 heads in 16 tosses is:$$P(X = 12) = \binom{16}{12}\left(\frac{1}{2}\right)^{16} = 0.0938$$The probability of getting 30 heads in 40 tosses is:$$P(X = 30) = \binom{40}{30}\left(\frac{1}{2}\right)^{40} = 0.0002$$The probability of getting 75 heads in 100 tosses is:$$P(X = 75) = \binom{100}{75}\left(\frac{1}{2}\right)^{100} = 0.0000$$Therefore, the least likely outcome is getting 75 heads in 100 tosses.

If a Coin Is Balanced So That P(Heads)= P(Tails)= 1/2,then