Calculating the Probability of Rolling Specified Numbers on a Die
To solve the problem, we first need to identify the divisors of 40 and the factors of 72, and then calculate the probability of rolling each.
The divisors of 40 (numbers that can divide 40 without leaving a remainder) are:
1, 2, 4, 5, 8, 10, 20, and 40.
If we assume the die is a standard six-sided die, then the only divisors of 40 that are possible to roll are 1, 2, 4, and 5. Thus, there are 4 favorable outcomes out of 6 possible outcomes when rolling a divisor of 40.
Now let's find the factors of 72 (numbers that 72 can be divided by without leaving a remainder), which are:
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.
Since a standard die only has six sides, the only factors of 72 that can appear on a die roll are 1, 2, 3, 4, and 6. This means there are 5 favorable outcomes out of 6 possible outcomes when rolling a factor of 72.
Now, to find the combined probability of these two independent events, we multiply the probabilities of each event occurring:
Probability of rolling a divisor of 40: 4/6 (because there are 4 possible divisors on a die out of 6 sides)
Probability of rolling a factor of 72: 5/6 (because there are 5 possible factors on a die out of 6 sides)
Combined probability = (4/6) * (5/6)
Let's simplify this fraction:
(4 * 5) / (6 * 6) = 20 / 36
Now we simplify the fraction by dividing both the numerator and the denominator by the greatest common divisor, which is 4:
20 / 36 = (20÷4) / (36÷4) = 5 / 9
So the probability of rolling a divisor of 40 and then rolling a factor of 72 is 5/9 when using a standard six-sided die.
