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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"

5 answers

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:

\[ \boxed{0.1250} \]

Assuming the outcomes to be equally likely, find the probability that the tosses are all the same.

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:

\[ \boxed{0.2500} \]

Assuming the outcomes to be equally likely, find the probability that exactly one of the three tosses is "Head."

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:

\[ \boxed{0.3750} \]