Question

a coin is tossed three times. an outcome is represented by a string of the sort htt (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.
table with outcomes: thh, hhh, hht, tht, hth, ttt, tth, htt; and events a, b, c with checkboxes and probability boxes

Explanation:

Event A: Two or more heads

Step 1: Identify outcomes with two or more heads

• THH: 2 heads (T, H, H) – include
• HHH: 3 heads – include
• HHT: 2 heads (H, H, T) – include
• THT: 1 head – exclude
• HTH: 2 heads (H, T, H) – include
• TTT: 0 heads – exclude
• TTH: 1 head – exclude
• HTT: 1 head – exclude

So the outcomes for Event A are THH, HHH, HHT, HTH.

Step 2: Calculate the number of favorable outcomes

Number of favorable outcomes, \( n(A) = 4 \)

Step 3: Calculate the probability

Total number of outcomes, \( N = 8 \)
Probability of Event A, \( P(A) = \frac{n(A)}{N} = \frac{4}{8} = \frac{1}{2} \)

Event B: A head on each of the last two tosses

Step 1: Identify outcomes with heads on last two tosses

• THH: last two tosses H, H – include
• HHH: last two tosses H, H – include
• HHT: last two tosses H, T – exclude
• THT: last two tosses H, T – exclude
• HTH: last two tosses T, H – exclude
• TTT: last two tosses T, T – exclude
• TTH: last two tosses T, H – exclude
• HTT: last two tosses T, T – exclude

So the outcomes for Event B are THH, HHH.

Step 2: Calculate the number of favorable outcomes

Number of favorable outcomes, \( n(B) = 2 \)

Step 3: Calculate the probability

Total number of outcomes, \( N = 8 \)
Probability of Event B, \( P(B) = \frac{n(B)}{N} = \frac{2}{8} = \frac{1}{4} \)

Event C: A tail on the first toss

Step 1: Identify outcomes with tail on first toss

• THH: first toss T – include
• HHH: first toss H – exclude
• HHT: first toss H – exclude
• THT: first toss T – include
• HTH: first toss H – exclude
• TTT: first toss T – include
• TTH: first toss T – include
• HTT: first toss H – exclude

So the outcomes for Event C are THH, THT, TTT, TTH.

Step 2: Calculate the number of favorable outcomes

Number of favorable outcomes, \( n(C) = 4 \)

Step 3: Calculate the probability

Total number of outcomes, \( N = 8 \)
Probability of Event C, \( P(C) = \frac{n(C)}{N} = \frac{4}{8} = \frac{1}{2} \)

Final Answers:

• Event A Probability: \(\boldsymbol{\frac{1}{2}}\)
• Event B Probability: \(\boldsymbol{\frac{1}{4}}\)
• Event C Probability: \(\boldsymbol{\frac{1}{2}}\)

Answer:

Event A: Two or more heads

Step 1: Identify outcomes with two or more heads

• THH: 2 heads (T, H, H) – include
• HHH: 3 heads – include
• HHT: 2 heads (H, H, T) – include
• THT: 1 head – exclude
• HTH: 2 heads (H, T, H) – include
• TTT: 0 heads – exclude
• TTH: 1 head – exclude
• HTT: 1 head – exclude

So the outcomes for Event A are THH, HHH, HHT, HTH.

Step 2: Calculate the number of favorable outcomes

Number of favorable outcomes, \( n(A) = 4 \)

Step 3: Calculate the probability

Total number of outcomes, \( N = 8 \)
Probability of Event A, \( P(A) = \frac{n(A)}{N} = \frac{4}{8} = \frac{1}{2} \)

Event B: A head on each of the last two tosses

Step 1: Identify outcomes with heads on last two tosses

• THH: last two tosses H, H – include
• HHH: last two tosses H, H – include
• HHT: last two tosses H, T – exclude
• THT: last two tosses H, T – exclude
• HTH: last two tosses T, H – exclude
• TTT: last two tosses T, T – exclude
• TTH: last two tosses T, H – exclude
• HTT: last two tosses T, T – exclude

So the outcomes for Event B are THH, HHH.

Step 2: Calculate the number of favorable outcomes

Number of favorable outcomes, \( n(B) = 2 \)

Step 3: Calculate the probability

Total number of outcomes, \( N = 8 \)
Probability of Event B, \( P(B) = \frac{n(B)}{N} = \frac{2}{8} = \frac{1}{4} \)

Event C: A tail on the first toss

Step 1: Identify outcomes with tail on first toss

• THH: first toss T – include
• HHH: first toss H – exclude
• HHT: first toss H – exclude
• THT: first toss T – include
• HTH: first toss H – exclude
• TTT: first toss T – include
• TTH: first toss T – include
• HTT: first toss H – exclude

So the outcomes for Event C are THH, THT, TTT, TTH.

Step 2: Calculate the number of favorable outcomes

Number of favorable outcomes, \( n(C) = 4 \)

Step 3: Calculate the probability

Total number of outcomes, \( N = 8 \)
Probability of Event C, \( P(C) = \frac{n(C)}{N} = \frac{4}{8} = \frac{1}{2} \)

Final Answers:

• Event A Probability: \(\boldsymbol{\frac{1}{2}}\)
• Event B Probability: \(\boldsymbol{\frac{1}{4}}\)
• Event C Probability: \(\boldsymbol{\frac{1}{2}}\)