Two fair dice are rolled 5 times and the sum of the numbers that come up is recorded. Find the probability of these events.
a. The sum is 7 on each of the five rolls.
b. The sum is 7 exactly four times in the five rolls.

When two fair dice are rolled, the possible sums range from 2 to 12, with some sums being more common than others. The sum of 7 is one of the most likely outcomes for two dice, as it can be achieved in 6 different combinations (1+6, 2+5, 3+4, 4+3, 5+2, and 6+1) out of a total of 36 possible combinations when rolling the dice. This gives a probability of rolling a sum of 7 as 6/36, which simplifies to 1/6.

To find the probability of rolling a sum of 7 on each of the five rolls, we use the formula for independent events. The probability of getting a sum of 7 on one roll is 1/6, so for five rolls, we compute (1/6)⁵. This yields a very small probability of (1/7776), which is equivalent to about 0.0129%.

Now, for getting a sum of 7 exactly four times in five rolls, we can apply the binomial probability formula. Here, we identify that there are 5 rolls, the number of successes (sum of 7) we want is 4, and the failure (not getting 7) in a single roll occurs with a probability of 5/6. Calculating it gives us:

This gives us a probability of 25/7776, around 0.00322, illustrating how getting close to perfection in probability must deal with that elusive, sneaky outcome!