Question

section 5.3 homework score: 2/13 answered: 2/13 question 3 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 definition

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 heads in a single toss is $P = \frac{1}{2}$.

Step2: Calculate probabilities for part (a)

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

Step3: Calculate probability for part (b)

We already found that the probability of getting heads on both tosses, which is $P(E\cap F)$. Since the sample space of two - coin tosses has $n(S) = 4$ elements and the event of getting heads on both tosses has $n(E\cap F)=1$ element, and using the formula $P=\frac{n(A)}{n(S)}$, we have $P(E\cap F)=\frac{1}{4}$.

Answer:

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