Question

test 2
score: 5/10 answered: 5/10
question 6
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?
input yes or no here:
(b) find the probability of showing heads on both tosses.
input your answer here:
question help: video

Explanation:

Step1: Check independence

For two events \(E\) and \(F\), if \(P(E\cap F)=P(E)\times P(F)\), they are independent.
When tossing a fair coin, \(P(E)=\frac{1}{2}\) (probability of heads on first toss), \(P(F)=\frac{1}{2}\) (probability of heads on second toss).
The sample space for two - coin tosses is \(\{HH, HT, TH, TT\}\), and \(P(E\cap F)=\frac{1}{4}\) (the event \(HH\)).
Since \(\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}\), \(E\) and \(F\) are independent.

Step2: Calculate \(P(E\cap F)\)

Using the formula for independent events \(P(E\cap F)=P(E)\times P(F)\).
We know \(P(E)=\frac{1}{2}\) and \(P(F)=\frac{1}{2}\).
So \(P(E\cap F)=\frac{1}{2}\times\frac{1}{2}\)

Answer:

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