Coin Flip Outcome Probability

Solution & Explanation

To solve this problem, we need to determine the probability that when flipping a randomly chosen coin twice, both flips result in the same face. We have two coins: a fair coin and a biased coin.

Step 1: Define the Probabilities for Each Coin

• Fair Coin:

  • Probability of heads (H): 1/2
  • Probability of tails (T): 1/2

• Biased Coin:

  • Probability of heads (H): 3/4
  • Probability of tails (T): 1/4

Step 2: Calculate Probability of Same Face for Each Coin

• For the Fair Coin:

  • Probability of two heads (HH): $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
  • Probability of two tails (TT): $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
  • Total probability of same face: $\frac{1}{4} + \frac{1}{4} = \frac{1}{2}$

• For the Biased Coin:

  • Probability of two heads (HH): $\frac{3}{4} \times \frac{3}{4} = \frac{9}{16}$
  • Probability of two tails (TT): $\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$
  • Total probability of same face: $\frac{9}{16} + \frac{1}{16} = \frac{10}{16} = \frac{5}{8}$

Step 3: Consider the Probability of Choosing Each Coin

Since the choice of coin is random, each coin has a probability of 1/2 of being chosen.

Step 4: Calculate Overall Probability

• Probability of same face with fair coin: $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
• Probability of same face with biased coin: $\frac{1}{2} \times \frac{5}{8} = \frac{5}{16}$

Add these probabilities together to find the total probability of the same face:

$\frac{1}{4} + \frac{5}{16} = \frac{4}{16} + \frac{5}{16} = \frac{9}{16}$

Thus, the probability that both flips result in the same face is $\frac{9}{16}$ or 0.5625.