Final answer:
The probability that a randomly selected coin from the box shows heads, given that there are 10% fair coins and 90% two-tailed coins, is calculated to be 5%.
Step-by-step explanation:
The probability that a randomly selected coin shows heads from a box with 10% fair coins and 90% two-tailed coins can be calculated using probability rules. First, we find the probability of selecting a fair coin and getting a head (P(Fair Coin and Head)), and then the probability of selecting a two-tailed coin and getting a head (P(Two-Tailed Coin and Head)), which is zero since a two-tailed coin cannot show a head. Since the fair coin has two sides, the probability of getting heads from it is 50%, or 0.5. So, we multiply the probability of picking a fair coin (10% or 0.10) by the probability of getting heads from a fair coin (0.5).
The complete calculation is:
- P(Fair Coin and Head) = P(Fair Coin) × P(Head | Fair Coin) = 0.10 × 0.5 = 0.05
- P(Two-Tailed Coin and Head) = P(Two-Tailed Coin) × P(Head | Two-Tailed Coin) = 0.90 × 0 = 0
Add the probabilities together since these are mutually exclusive events (a coin cannot be both fair and two-tailed) to find the total probability:
P(Head) = P(Fair Coin and Head) + P(Two-Tailed Coin and Head) = 0.05 + 0 = 0.05.
So, the probability that a randomly selected coin from the box shows heads is 5%.