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A fair coin is tossed three times, and the events A and B
Chapter 3, Problem 29E(choose chapter or problem)
A fair coin is tossed three times, and the events A and B are defined as follows:
A: {At least one head is observed.}
B: {The number of heads observed is odd.}
a. Identify the sample points in the events \(A, B, A \cup B, A^{c}\), and \(A \cap B\).
b. Find \(P(A), P(B), P(A \cup B), P\left(A^{c}\right)\), and \(P(A \cap B)\) by summing the probabilities of the appropriate sample points.
c. Find \(P(A \cup B)\) using the additive rule. Compare your answer to the one you obtained in part b.
d. Are the events A and B mutually exclusive? Why?
Questions & Answers
QUESTION:
A fair coin is tossed three times, and the events A and B are defined as follows:
A: {At least one head is observed.}
B: {The number of heads observed is odd.}
a. Identify the sample points in the events \(A, B, A \cup B, A^{c}\), and \(A \cap B\).
b. Find \(P(A), P(B), P(A \cup B), P\left(A^{c}\right)\), and \(P(A \cap B)\) by summing the probabilities of the appropriate sample points.
c. Find \(P(A \cup B)\) using the additive rule. Compare your answer to the one you obtained in part b.
d. Are the events A and B mutually exclusive? Why?
ANSWER:Step 1 of 4
Given that a fair coin is tossed three times then the sample space is
S = {(HHH), (HHT), (HTH), (HTT), (THH), (THT), (TTH), (TTT)}
a) We have to write the sample points of \(\mathbf{A}, \mathbf{B}, \mathbf{A U B}, \mathbf{A}^{\mathbf{C}}\), and \(\mathbf{A} \cap \mathbf{B}\)
Now A:{At least one head is observed}
Then A = {(HHH), (HHT), (HTH), (THH), (THT), (TTH)}
And B:{The no.of heads observed is odd}
The B = {(HHH), (HTT), (THT), (TTH)}
Now AUB = {(HHH), (HHT), (HTH), (THH), (THT), (TTH)}
Now \(\mathrm{A}^{\mathrm{C}}=\{(\mathrm{TTT})\}\)
And \(\mathrm{A} \cap \mathrm{B}=\{(\mathrm{HHH}),(\mathrm{HTT}),(\mathrm{THT}),(\mathrm{TTH})\}\)