Question

a fair coin is tossed three times in succession. the set of equally likely outcomes is {hhh, hht, hth, thh, htt, tht, tth, ttt}. find the probability of getting exactly two heads. the probability of getting two heads is □. (type an integer or a simplified fraction.)

Explanation:

Step1: Determine total number of outcomes

When a coin is tossed \(n = 3\) times, and each toss has \(2\) possible outcomes (head \(H\) or tail \(T\)), the total number of outcomes \(N\) is \(N=2\times2\times2=2^{3}=8\) (using the counting principle \(N = 2^{n}\) for \(n\) coin - tosses).

Step2: Determine number of favorable outcomes

We use the combination formula \(C(n,k)=\frac{n!}{k!(n - k)!}\), where \(n = 3\) (number of coin - tosses) and \(k = 2\) (number of heads). \(C(3,2)=\frac{3!}{2!(3 - 2)!}=\frac{3!}{2!1!}=\frac{3\times2!}{2!×1}=3\). The favorable outcomes are \(\{HHT, HTH, THH\}\).

Step3: Calculate the probability

The probability \(P\) of an event is given by \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\). So \(P=\frac{3}{8}\).

Answer:

\(\frac{3}{8}\)