Coins and Probability

Solution & Explanation

To solve this problem, we will use Bayes' Theorem to calculate the probability that the chosen coin is the double-headed one, given that we observed ten consecutive heads. Then, we'll determine the probability that the next flip will also result in a head.

Step 1: Define the Events

• F: The event that the coin is fair.
• Fᶜ: The event that the coin is double-headed.
• H=10: The event that ten consecutive heads are observed.

Step 2: Calculate Prior Probabilities

• P(F): Probability of selecting a fair coin = 999/1000 = 0.999.
• P(Fᶜ): Probability of selecting a double-headed coin = 1/1000 = 0.001.

Step 3: Calculate Event Probabilities

• P(H=10 | F): Probability of observing ten heads with a fair coin = (0.5)^10 ≈ 0.0009765625.
• P(H=10 | Fᶜ): Probability of observing ten heads with a double-headed coin = 1 (since all flips will be heads).

Step 4: Apply Bayes' Theorem

Bayes' Theorem states:

$P(Fᶜ | H=10) = \frac{P(H=10 | Fᶜ) \cdot P(Fᶜ)}{P(H=10)}$

Calculate the Denominator, P(H=10):

$P(H=10) = P(H=10 | F) \cdot P(F) + P(H=10 | Fᶜ) \cdot P(Fᶜ)$

Substitute the values:

$P(H=10) = (0.0009765625 \cdot 0.999) + (1 \cdot 0.001) ≈ 0.0019755859375$

Calculate P(Fᶜ | H=10):

$P(Fᶜ | H=10) = \frac{1 \cdot 0.001}{0.0019755859375} ≈ 0.506$

Thus, the probability that the chosen coin is the double-headed one given ten consecutive heads is approximately 50.6%.

Step 5: Determine Probability of Next Flip Resulting in a Head

Using the law of total probability, we calculate:

$P(H_{next}) = P(H_{next} | Fᶜ) \cdot P(Fᶜ | H=10) + P(H_{next} | F) \cdot P(F | H=10)$

Where:

• P(H_{next} | Fᶜ) = 1 (since it's double-headed).
• P(H_{next} | F) = 0.5 (since it's a fair coin).
• P(F | H=10) = 1 - P(Fᶜ | H=10) ≈ 0.494.

Substitute the values:

$P(H_{next}) = (1 \cdot 0.506) + (0.5 \cdot 0.494) ≈ 0.753$

Therefore, the probability that the next flip will result in a head is approximately 75.3%.