Question

a coin is tossed three times. an outcome is represented by a string of the sort hit (meaning a head on the first toss, followed by two tails). the 8 outcomes are listed in the table below. note that each outcome has the same probability. for each of the three events in the table, check the outcome(s) that are contained in the event. then, in the last column, enter the probability of the event. event a: a tail on the last toss event b: a head on each of the first two tosses event c: no tails on the first two tosses outcomes hth tht thh hht htt ttt tth hhh probability

Explanation:

Step1: Analyze Event A

Event A: A tail on the last toss.
For HTH: last toss is H (no)
For THT: last toss is T (yes)
For THH: last toss is H (no)
For HHT: last toss is T (yes)
For HTT: last toss is T (yes)
For TTT: last toss is T (yes)
For TTH: last toss is H (no)
For HHH: last toss is H (no)
Number of outcomes in Event A: 4 (THT, HHT, HTT, TTT)
Probability of each outcome: \(\frac{1}{8}\)
Probability of Event A: \(4\times\frac{1}{8}=\frac{1}{2}\)

Step2: Analyze Event B

Event B: A head on each of the first two tosses.
For HTH: first two tosses H and T (no)
For THT: first two tosses T and H (no)
For THH: first two tosses T and H (no)
For HHT: first two tosses H and H (yes)
For HTT: first two tosses H and T (no)
For TTT: first two tosses T and T (no)
For TTH: first two tosses T and T (no)
For HHH: first two tosses H and H (yes)
Number of outcomes in Event B: 2 (HHT, HHH)
Probability of Event B: \(2\times\frac{1}{8}=\frac{1}{4}\)

Step3: Analyze Event C

Event C: No tails on the first two tosses.
For HTH: first two tosses H and T (no)
For THT: first two tosses T and H (no)
For THH: first two tosses T and H (no)
For HHT: first two tosses H and H (yes)
For HTT: first two tosses H and T (no)
For TTT: first two tosses T and T (no)
For TTH: first two tosses T and T (no)
For HHH: first two tosses H and H (yes)
Number of outcomes in Event C: 2 (HHT, HHH)
Probability of Event C: \(2\times\frac{1}{8}=\frac{1}{4}\)

Answer:

Outcome	Event A (Tail on last toss)	Event B (Head on first two tosses)	Event C (No tails on first two tosses)
THT	✔️
THH
HHT	✔️	✔️	✔️
HTT	✔️
TTT	✔️
TTH
HHH		✔️	✔️
Probability	\(\frac{1}{2}\)	\(\frac{1}{4}\)	\(\frac{1}{4}\)