Question
A coin flip: A fair coin is tossed three times. The outcomes of the three tosses are recorded. Round your answers to four decimal places if necessary.
Find the probability that all 3 tosses are "tails"
Find the probability that all 3 tosses are "tails"
Answers
GPT-4o mini
To find the probability that all three tosses of a fair coin result in "tails," we can use the concept of probability for independent events.
1. **Probability of tails on one flip**: Since the coin is fair, the probability of getting tails on one flip is:
\[
P(Tails) = \frac{1}{2}
\]
2. **Probability of tails on three flips**: Since the flips are independent, the probability of getting tails on all three flips is:
\[
P(All \, Tails) = P(Tails) \times P(Tails) \times P(Tails) = \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^3 = \frac{1}{8}
\]
3. **Convert to decimal form**:
\[
\frac{1}{8} = 0.125
\]
Thus, the probability that all three tosses result in tails is:
\[
\boxed{0.1250}
\]
1. **Probability of tails on one flip**: Since the coin is fair, the probability of getting tails on one flip is:
\[
P(Tails) = \frac{1}{2}
\]
2. **Probability of tails on three flips**: Since the flips are independent, the probability of getting tails on all three flips is:
\[
P(All \, Tails) = P(Tails) \times P(Tails) \times P(Tails) = \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^3 = \frac{1}{8}
\]
3. **Convert to decimal form**:
\[
\frac{1}{8} = 0.125
\]
Thus, the probability that all three tosses result in tails is:
\[
\boxed{0.1250}
\]
Assuming the outcomes to be equally likely, find the probability that the tosses are all the same.
GPT-4o mini
To find the probability that all three tosses of a fair coin are the same (either all heads or all tails), we can first determine the total number of outcomes and the number of favorable outcomes.
1. **Total Outcomes**: When a coin is tossed three times, each toss has 2 possible outcomes (Heads or Tails). Therefore, the total number of outcomes for three tosses is:
\[
2^3 = 8
\]
The possible outcomes are:
- HHH (all heads)
- HHT
- HTH
- HTT
- THH
- THT
- TTH
- TTT (all tails)
2. **Favorable Outcomes**: The outcomes in which all tosses result in the same are:
- HHH
- TTT
Thus, there are 2 favorable outcomes.
3. **Calculate Probability**: The probability that all tosses are the same is given by the ratio of favorable outcomes to total outcomes:
\[
P(All \, Same) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} = \frac{2}{8} = \frac{1}{4}
\]
4. **Convert to Decimal Form**:
\[
\frac{1}{4} = 0.25
\]
Thus, the probability that all tosses are the same is:
\[
\boxed{0.2500}
\]
1. **Total Outcomes**: When a coin is tossed three times, each toss has 2 possible outcomes (Heads or Tails). Therefore, the total number of outcomes for three tosses is:
\[
2^3 = 8
\]
The possible outcomes are:
- HHH (all heads)
- HHT
- HTH
- HTT
- THH
- THT
- TTH
- TTT (all tails)
2. **Favorable Outcomes**: The outcomes in which all tosses result in the same are:
- HHH
- TTT
Thus, there are 2 favorable outcomes.
3. **Calculate Probability**: The probability that all tosses are the same is given by the ratio of favorable outcomes to total outcomes:
\[
P(All \, Same) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} = \frac{2}{8} = \frac{1}{4}
\]
4. **Convert to Decimal Form**:
\[
\frac{1}{4} = 0.25
\]
Thus, the probability that all tosses are the same is:
\[
\boxed{0.2500}
\]
Assuming the outcomes to be equally likely, find the probability that exactly one of the three tosses is "Head."
GPT-4o mini
To find the probability that exactly one of the three tosses of a fair coin results in a "Head," we will follow these steps:
1. **Determine the Total Outcomes**: As previously mentioned, when a coin is tossed three times, the total number of outcomes is:
\[
2^3 = 8
\]
2. **Identify Favorable Outcomes**: We need to identify the outcomes in which exactly one toss results in Heads. The possible arrangements for exactly one Head (H) and two Tails (T) are:
- HTT
- THT
- TTH
Thus, there are 3 favorable outcomes.
3. **Calculate the Probability**: The probability that exactly one of the three tosses is Heads is given by the ratio of favorable outcomes to the total outcomes:
\[
P(Exactly \, One \, Head) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} = \frac{3}{8}
\]
4. **Convert to Decimal Form**:
\[
\frac{3}{8} = 0.375
\]
Thus, the probability that exactly one of the three tosses is Heads is:
\[
\boxed{0.3750}
\]
1. **Determine the Total Outcomes**: As previously mentioned, when a coin is tossed three times, the total number of outcomes is:
\[
2^3 = 8
\]
2. **Identify Favorable Outcomes**: We need to identify the outcomes in which exactly one toss results in Heads. The possible arrangements for exactly one Head (H) and two Tails (T) are:
- HTT
- THT
- TTH
Thus, there are 3 favorable outcomes.
3. **Calculate the Probability**: The probability that exactly one of the three tosses is Heads is given by the ratio of favorable outcomes to the total outcomes:
\[
P(Exactly \, One \, Head) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} = \frac{3}{8}
\]
4. **Convert to Decimal Form**:
\[
\frac{3}{8} = 0.375
\]
Thus, the probability that exactly one of the three tosses is Heads is:
\[
\boxed{0.3750}
\]