Questions: 19. Heads or Taits Three ordinary quarters and a fake quarter with two heads are placed in a hat. One quarter is selected at random and tossed twice. If the outcome is "HH," what is the probability that the fake quarter was selected?

Solution Steps

To solve this problem, we use Bayes' Theorem. We need to find the probability that the fake quarter was selected given that the outcome is "HH". First, calculate the probability of getting "HH" with each type of quarter. Then, use these probabilities to apply Bayes' Theorem.

We need to find the probability that the fake quarter was selected given that the outcome of tossing the quarter twice is "HH".

• Probability of selecting the fake quarter: \[ P(\text{Fake}) = \frac{1}{4} = 0.25 \]

• Probability of selecting a real quarter: \[ P(\text{Real}) = \frac{3}{4} = 0.75 \]

• Probability of "HH" with the fake quarter (always heads): \[ P(\text{HH} \mid \text{Fake}) = 1 \]

• Probability of "HH" with a real quarter (each toss is independent with probability \(\frac{1}{2}\) for heads): \[ P(\text{HH} \mid \text{Real}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} = 0.25 \]

Using the law of total probability: \[ P(\text{HH}) = P(\text{HH} \mid \text{Fake}) \cdot P(\text{Fake}) + P(\text{HH} \mid \text{Real}) \cdot P(\text{Real}) \] \[ P(\text{HH}) = 1 \cdot 0.25 + 0.25 \cdot 0.75 = 0.25 + 0.1875 = 0.4375 \]

Using Bayes' Theorem to find \( P(\text{Fake} \mid \text{HH}) \): \[ P(\text{Fake} \mid \text{HH}) = \frac{P(\text{HH} \mid \text{Fake}) \cdot P(\text{Fake})}{P(\text{HH})} \] \[ P(\text{Fake} \mid \text{HH}) = \frac{1 \cdot 0.25}{0.4375} \approx 0.5714 \]

Final Answer