Question

a coin will be tossed twice. let e be the event \the first toss shows heads\ and f the event \the second toss shows heads\. (a) are the events e and f independent? yes no (b) find the probability of showing heads on both tosses.

Explanation:

Step1: Recall independence concept

Two events $E$ and $F$ are independent if $P(E\cap F)=P(E)\times P(F)$. For a fair - coin toss, the probability of getting a head in a single toss is $P = \frac{1}{2}$. The probability of event $E$ (first toss is head) is $P(E)=\frac{1}{2}$, and the probability of event $F$ (second toss is head) is $P(F)=\frac{1}{2}$. The sample space of two - coin tosses is $S=\{HH, HT, TH, TT\}$, and $E\cap F=\{HH\}$, so $P(E\cap F)=\frac{1}{4}$. Also, $P(E)\times P(F)=\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$. Since $P(E\cap F) = P(E)\times P(F)$, the events are independent.

Step2: Calculate probability of both heads

The probability of getting a head on the first toss is $P_1=\frac{1}{2}$, and the probability of getting a head on the second toss is $P_2=\frac{1}{2}$. Since the two coin - tosses are independent events, the probability of getting heads on both tosses is $P = P_1\times P_2$. Using the multiplication rule for independent events, $P=\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$.

Answer:

(a) Yes
(b) $\frac{1}{4}$