To calculate the probability of rolling an even number on a six-sided die and flipping heads simultaneously, we can approach this problem in two parts: the probability of each individual event and then combining them.
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Probability of rolling an even number on a six-sided die: The even numbers on a six-sided die are 2, 4, and 6. There are 3 even numbers out of a total of 6 possible outcomes (1, 2, 3, 4, 5, 6).
\[ P(\text{even number}) = \frac{3 \text{ even outcomes}}{6 \text{ total outcomes}} = \frac{1}{2} \]
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Probability of flipping heads on a coin: When flipping a fair coin, there are 2 outcomes (heads or tails), so the probability of flipping heads is:
\[ P(\text{heads}) = \frac{1 \text{ heads outcome}}{2 \text{ total outcomes}} = \frac{1}{2} \]
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Combining the probabilities: Since the roll of the die and the coin flip are independent events, we can multiply the probabilities of the two events to find the combined probability of both occurring simultaneously:
\[ P(\text{even number and heads}) = P(\text{even number}) \times P(\text{heads}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \]
Thus, the probability of rolling an even number on a six-sided die and flipping heads simultaneously is \( \frac{1}{4} \).