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If you roll two fair six-sided dice, what is the probability that both dice show an odd number?

If you roll two fair six-sided dice, what is the probability that both dice show an odd number?

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Outcomes of compound events

Full solution

Q. If you roll two fair six-sided dice, what is the probability that both dice show an odd number?

Identify Odd Outcomes: First, figure out the number of odd outcomes on one die. Odd numbers on a die are $1$ , $3$ , and $5$ .
Calculate Probability of Odd Number on One Die: So, there are $3$ odd outcomes out of $6$ total outcomes on one die. The probability of rolling an odd number on one die is $\frac{3}{6}$ , which simplifies to $\frac{1}{2}$ .
Multiply Probabilities for Combined Probability: Now, since the two dice are independent, multiply the probabilities of each die to get the combined probability. So, $(\frac{1}{2}) \times (\frac{1}{2}) = \frac{1}{4}$ .
Correct Calculation: But wait, I made a mistake. The probability of rolling an odd number on one die is actually $\frac{3}{6}$ , which doesn't simplify to $\frac{1}{2}$ , it simplifies to $\frac{1}{2}$ . So the correct calculation should be $(\frac{1}{2}) \times (\frac{1}{2}) = \frac{1}{4}$ .

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