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.
event a: no tails on the first two tosses
event b: a tail on the first toss or the second toss (or both)
event c: a tail on the first toss

Explanation:

To solve this, we analyze each event by identifying the relevant outcomes and calculating their probabilities. Each toss is independent, and there are \( 2^3 = 8 \) total outcomes, each with probability \( \frac{1}{8} \).

Event A: No tails on the first two tosses

• Step 1: Identify outcomes

The first two tosses must be heads (H). The third toss can be H or T.
Outcomes: \( \text{HHH} \), \( \text{HHT} \) (2 outcomes).

• Step 2: Calculate probability

Probability = \( \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{2}{8} = \frac{1}{4} \).

Event B: A tail on the first toss or the second toss (or both)

• Step 1: Identify outcomes

Use the complement: “No tail on first AND no tail on second” (i.e., first two tosses are heads) has outcomes \( \text{HHH} \), \( \text{HHT} \) (2 outcomes).
Thus, the complement of Event B has 2 outcomes.
Favorable outcomes for B = Total outcomes − Complement outcomes = \( 8 - 2 = 6 \).
Outcomes: \( \text{THH} \), \( \text{THT} \), \( \text{TTH} \), \( \text{TTT} \), \( \text{HTT} \), \( \text{HTH} \) (6 outcomes).

• Step 2: Calculate probability

Probability = \( \frac{6}{8} = \frac{3}{4} \).

Event C: A tail on the first toss

• Step 1: Identify outcomes

The first toss is T. The second and third tosses can be any (H or T).
Outcomes: \( \text{THH} \), \( \text{THT} \), \( \text{TTH} \), \( \text{TTT} \) (4 outcomes).

• Step 2: Calculate probability

Probability = \( \frac{4}{8} = \frac{1}{2} \).

Final Answers

• Event A: \( \boldsymbol{\frac{1}{4}} \)
• Event B: \( \boldsymbol{\frac{3}{4}} \)
• Event C: \( \boldsymbol{\frac{1}{2}} \)

(For the table, check the boxes for the identified outcomes and enter the probabilities as above.)

Answer:

To solve this, we analyze each event by identifying the relevant outcomes and calculating their probabilities. Each toss is independent, and there are \( 2^3 = 8 \) total outcomes, each with probability \( \frac{1}{8} \).

Event A: No tails on the first two tosses

• Step 1: Identify outcomes

The first two tosses must be heads (H). The third toss can be H or T.
Outcomes: \( \text{HHH} \), \( \text{HHT} \) (2 outcomes).

• Step 2: Calculate probability

Probability = \( \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{2}{8} = \frac{1}{4} \).

Event B: A tail on the first toss or the second toss (or both)

• Step 1: Identify outcomes

Use the complement: “No tail on first AND no tail on second” (i.e., first two tosses are heads) has outcomes \( \text{HHH} \), \( \text{HHT} \) (2 outcomes).
Thus, the complement of Event B has 2 outcomes.
Favorable outcomes for B = Total outcomes − Complement outcomes = \( 8 - 2 = 6 \).
Outcomes: \( \text{THH} \), \( \text{THT} \), \( \text{TTH} \), \( \text{TTT} \), \( \text{HTT} \), \( \text{HTH} \) (6 outcomes).

• Step 2: Calculate probability

Probability = \( \frac{6}{8} = \frac{3}{4} \).

Event C: A tail on the first toss

• Step 1: Identify outcomes

The first toss is T. The second and third tosses can be any (H or T).
Outcomes: \( \text{THH} \), \( \text{THT} \), \( \text{TTH} \), \( \text{TTT} \) (4 outcomes).

• Step 2: Calculate probability

Probability = \( \frac{4}{8} = \frac{1}{2} \).

Final Answers

• Event A: \( \boldsymbol{\frac{1}{4}} \)
• Event B: \( \boldsymbol{\frac{3}{4}} \)
• Event C: \( \boldsymbol{\frac{1}{2}} \)

(For the table, check the boxes for the identified outcomes and enter the probabilities as above.)