Suppose I have three coins in my pocket: the first lands heads with probability

Question

Suppose I have three coins in my pocket: the first lands heads with probability 0.1, the second with probability 0.5, and the third with probability 0.9. I select a coin at random from my pocket and toss it twice. Let Ci denote the event that I choose coin i, for i = 1.2.3. Also denote Hi as the event that the jth toss lands heads, for J-1, 2. Evaluate the following probabilities: a) P(Hi) Hint: calculate P(C1 N H), P(2 n Hi) and P(Csn Hi) first b) P(C H), PC2 H) and P(Cs | Hi). c) P(H2 H) Hint: you may want to rely on results in b.

Explanation / Answer

a) P(H1) = P( first coin and heads+second coin and heads+third coin and heads)

=P(C1)*P(H1|C1)+P(C2)*P(H1|C2)+P(C3)*P(H1|C3)

(1/3)*(0.1)+(1/3)*(0.5)+(1/3)*(0.9)=0.5

b) P(C1|H1) =P(C1)*P(H1|C1)/P(H1) = (1/3)*(0.1)/0.5=0.0667

P(C2|H1)=P(C2)*P(H1|C2)/P(H1)=(1/3)*(0.5)/0.5=0.3333

P(C3|H1)=P(C3)*P(H1|C3)/P(H1=(1/3)*(0.9)/0.5 =0.6

c) P(H1 n H2)=P(C1)*P(H1|C1)*P(H2|C1)+P(C2)*P(H1|C2)P(H2|C2)+P(C3)*P(H1|C3)*P(H2|C3)

=(1/3)*(0.1)*(0.1)+(1/3)*(0.5)*(0.5)+(1/3)*(0.9)*(0.9)=0.3567

hence P(H2|H1) =P(H1 nH2)/P(H1) =0.3567/0.5=0.7133

please revert for any clarification required.

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.