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To solve the problem of finding the probability for each given event when a coin is tossed three times, we should carefully analyze each event and determine the number of favorable outcomes. Here is the detailed step-by-step solution:
### Event A: Two or more tails
1. Identify outcomes with two or more tails:
- HHH: 0 tails
- HTH: 1 tail
- HHT: 1 tail
- HTT: 2 tails (favorable)
- TTH: 2 tails (favorable)
- THT: 2 tails (favorable)
- TTT: 3 tails (favorable)
- THH: 1 tail
2. Favorable outcomes for Event A:
HTT, TTH, THT, TTT
3. Number of favorable outcomes for Event A: 4
4. Probability of Event A:
Since there are 8 possible outcomes and 4 favorable ones:
[tex]\( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{4}{8} = 0.5 \)[/tex]
### Event B: A head on each of the last two tosses
1. Identify outcomes with a head on each of the last two tosses:
- HHH: last two tosses are HH (favorable)
- HTH: last two tosses are TH
- HHT: last two tosses are HT
- HTT: last two tosses are TT
- TTH: last two tosses are TH
- THT: last two tosses are HT
- TTT: last two tosses are TT
- THH: last two tosses are HH (favorable)
2. Favorable outcomes for Event B:
HHH, THH
3. Number of favorable outcomes for Event B: 2
4. Probability of Event B:
Since there are 8 possible outcomes and 2 favorable ones:
[tex]\( P(B) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{2}{8} = 0.25 \)[/tex]
### Event C: A tail on the last toss
1. Identify outcomes with a tail on the last toss:
- HHH: last toss is H
- HTH: last toss is H
- HHT: last toss is T (favorable)
- HTT: last toss is T (favorable)
- TTH: last toss is H
- THT: last toss is T (favorable)
- TTT: last toss is T (favorable)
- THH: last toss is H
2. Favorable outcomes for Event C:
HHT, HTT, THT, TTT
3. Number of favorable outcomes for Event C: 4
4. Probability of Event C:
Since there are 8 possible outcomes and 4 favorable ones:
[tex]\( P(C) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{4}{8} = 0.5 \)[/tex]
### Summary:
| Event | Favorable Outcomes | Number of Favorable Outcomes | Probability |
|-------|---------------------|------------------------------|-------------|
| A | HTT, TTH, THT, TTT | 4 | 0.5 |
| B | HHH, THH | 2 | 0.25 |
| C | HHT, HTT, THT, TTT | 4 | 0.5 |
Thus, the probabilities for Events A, B, and C are 0.5, 0.25, and 0.5, respectively.
### Event A: Two or more tails
1. Identify outcomes with two or more tails:
- HHH: 0 tails
- HTH: 1 tail
- HHT: 1 tail
- HTT: 2 tails (favorable)
- TTH: 2 tails (favorable)
- THT: 2 tails (favorable)
- TTT: 3 tails (favorable)
- THH: 1 tail
2. Favorable outcomes for Event A:
HTT, TTH, THT, TTT
3. Number of favorable outcomes for Event A: 4
4. Probability of Event A:
Since there are 8 possible outcomes and 4 favorable ones:
[tex]\( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{4}{8} = 0.5 \)[/tex]
### Event B: A head on each of the last two tosses
1. Identify outcomes with a head on each of the last two tosses:
- HHH: last two tosses are HH (favorable)
- HTH: last two tosses are TH
- HHT: last two tosses are HT
- HTT: last two tosses are TT
- TTH: last two tosses are TH
- THT: last two tosses are HT
- TTT: last two tosses are TT
- THH: last two tosses are HH (favorable)
2. Favorable outcomes for Event B:
HHH, THH
3. Number of favorable outcomes for Event B: 2
4. Probability of Event B:
Since there are 8 possible outcomes and 2 favorable ones:
[tex]\( P(B) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{2}{8} = 0.25 \)[/tex]
### Event C: A tail on the last toss
1. Identify outcomes with a tail on the last toss:
- HHH: last toss is H
- HTH: last toss is H
- HHT: last toss is T (favorable)
- HTT: last toss is T (favorable)
- TTH: last toss is H
- THT: last toss is T (favorable)
- TTT: last toss is T (favorable)
- THH: last toss is H
2. Favorable outcomes for Event C:
HHT, HTT, THT, TTT
3. Number of favorable outcomes for Event C: 4
4. Probability of Event C:
Since there are 8 possible outcomes and 4 favorable ones:
[tex]\( P(C) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{4}{8} = 0.5 \)[/tex]
### Summary:
| Event | Favorable Outcomes | Number of Favorable Outcomes | Probability |
|-------|---------------------|------------------------------|-------------|
| A | HTT, TTH, THT, TTT | 4 | 0.5 |
| B | HHH, THH | 2 | 0.25 |
| C | HHT, HTT, THT, TTT | 4 | 0.5 |
Thus, the probabilities for Events A, B, and C are 0.5, 0.25, and 0.5, respectively.
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