Question
What is the theoretical probability a black card is picked and the coin lands on heads? Type your answer as a fraction in simplest form.
Answers
GPT-4o mini
To calculate the theoretical probability of both events happening (picking a black card and the coin landing on heads), we first need to consider the individual probabilities of each event.
1. **Picking a black card**:
In a standard deck of 52 playing cards, there are 26 black cards (13 spades and 13 clubs). Thus, the probability of picking a black card is:
\[
P(\text{black card}) = \frac{26}{52} = \frac{1}{2}
\]
2. **Coin landing on heads**:
A standard coin has 2 sides: heads and tails. The probability of the coin landing on heads is:
\[
P(\text{heads}) = \frac{1}{2}
\]
Since these two events (picking a black card and flipping a coin) are independent, the combined probability of both events occurring is given by multiplying their individual probabilities:
\[
P(\text{black card and heads}) = P(\text{black card}) \times P(\text{heads}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}
\]
Thus, the theoretical probability that a black card is picked and the coin lands on heads is:
\[
\boxed{\frac{1}{4}}
\]
1. **Picking a black card**:
In a standard deck of 52 playing cards, there are 26 black cards (13 spades and 13 clubs). Thus, the probability of picking a black card is:
\[
P(\text{black card}) = \frac{26}{52} = \frac{1}{2}
\]
2. **Coin landing on heads**:
A standard coin has 2 sides: heads and tails. The probability of the coin landing on heads is:
\[
P(\text{heads}) = \frac{1}{2}
\]
Since these two events (picking a black card and flipping a coin) are independent, the combined probability of both events occurring is given by multiplying their individual probabilities:
\[
P(\text{black card and heads}) = P(\text{black card}) \times P(\text{heads}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}
\]
Thus, the theoretical probability that a black card is picked and the coin lands on heads is:
\[
\boxed{\frac{1}{4}}
\]