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Assume that a coin is tossed twice. The coin may not be fair. The sample space consists of the outcomes {HH, HT, TH, TT}.

outcome: HH , probability 0.03
outcome: HT, probability 0.66
outcome TH, probability -0.45
outcome: TT, probability -0.19
is the given a probability model?

1 answer

To determine whether the given set of outcomes and their associated probabilities forms a valid probability model, we need to check two conditions:

  1. The probabilities must be non-negative (i.e., greater than or equal to 0).
  2. The sum of all probabilities must equal 1.

Let's analyze the given probabilities:

  • Probability of HH: 0.03
  • Probability of HT: 0.66
  • Probability of TH: -0.45
  • Probability of TT: -0.19

Checking the non-negativity condition:

  • Probability of HH (0.03) is non-negative.
  • Probability of HT (0.66) is non-negative.
  • Probability of TH (-0.45) is negative.
  • Probability of TT (-0.19) is negative.

Since the probabilities for TH and TT are negative, the first condition is violated.

Checking the sum of probabilities:

  • \(0.03 + 0.66 - 0.45 - 0.19 = 0.03 + 0.66 - 0.45 - 0.19 = 0.05 \)

As a result, the sum of the probabilities does not equal 1.

Since both conditions for a valid probability model are not satisfied (due to negative probabilities and the sum not equaling 1), the given set of outcomes and probabilities does not constitute a valid probability model.