Question

A pair of dice is thrown 120 times. What is the approximate probability of throwing at least 15 sevens? Assume that the rolls are independent and remember that the probability of rolling a seven on a single roll is \(6 / 36=1 / 6\).

Short answer

Using the binomial probability formula, we can find the probability of throwing at least 15 sevens in 120 rolls. First, we need to calculate the combinations term C(n, k) for each k = 15, 16, ... 120 using the formula \(C(n, k) = \frac{n!}{k!(n-k)!}\). Then, calculate the probabilities for each k using the binomial probability formula \(P(X=k) = C(n,k) * p^k * (1-p)^{n-k}\), and sum the probabilities for each k = 15, 16, ... 120 to find the probability of getting at least 15 sevens: \(P(X\geq 15) = \sum_{k=15}^{120} P(X=k)\). A computer or calculator can be used to calculate the probabilities and sum them up to get the approximate probability.

Step-by-step solution

Set up the binomial probability formula

Using the binomial probability formula: \(P(X=k) = C(n,k) * p^k * (1-p)^{n-k}\) where: - n = 120, the number of trials - k = 15, 16, ..., 120, since we need the sum of probabilities of getting exactly these numbers of sevens - p = 1/6, the probability of rolling a seven

Calculate the combination term C(n, k)

We need to calculate C(n, k) for each k = 15, 16, ..., 120. Here, the combinations formula is: \(C(n, k) = \frac{n!}{k!(n-k)!}\)

Calculate the probabilities for each k

For each k = 15, 16, ..., 120, calculate the probability using the binomial probability formula: \(P(X=k) = C(n,k) * p^k * (1-p)^{n-k}\)

Sum the probabilities to find the probability of at least 15 sevens

Sum the probabilities for each k = 15, 16, ..., 120 to find the probability of getting at least 15 sevens: \(P(X\geq 15) = \sum_{k=15}^{120} P(X=k)\) Now, calculating the probabilities and summing them up might be time-consuming, so we can use a computer or calculator to help us with that. Once we have the sum, we can get the approximate probability of throwing at least 15 sevens.

Key concepts

Probability Theory

Probability theory is a branch of mathematics that deals with the likelihood of different outcomes occurring. An easy way of thinking about probability is considering it as a measure of how certain we can be that an event will happen. The probability of an event is a number between 0 and 1, where 0 means the event is impossible, and 1 means it is certain to occur.

In our exercise, the event is 'rolling a seven with a pair of dice'. Each throw of the dice is an independent trial, and we are trying to find out the likelihood of rolling at least 15 sevens out of 120 throws. The formula we use for such a case is part of probability theory and specifically taps into the rules surrounding independent events and fixed probabilities.

Binomial Distribution

The binomial distribution is a probability distribution that summarizes the likelihood that a value will take on one of two independent values under a given number of occurrences or trials. This type of distribution is relevant when there are exactly two mutually exclusive outcomes of a trial, often termed as 'success' and 'failure'.

For instance, when rolling dice, getting a seven can be seen as a success and not getting a seven as a failure. The binomial distribution requires two parameters: the number of trials (n) and the probability of success (p) on an individual trial. Our dice-rolling exercise involves computing these parameters into the binomial probability formula to find the likelihood of having a success (rolling a seven) at least 15 times in 120 trials.

Combinatorics

Combinatorics is a field of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is related to many other areas of mathematics, such as algebra, probability, and geometry, and is highly useful in practical applications such as computer science, statistics, and logistics.

In our problem, combinatory calculations help determine the number of possible outcomes in which you can obtain 15 sevens within 120 dice rolls. The combination formula, represented as C(n, k), allows us to compute the different ways we can achieve k successes (rolling sevens) in n trials (dice throws), without regard to the order of the successes. It simplifies our calculations when applying the binomial probability formula.

Independent Events

Independent events are a fundamental concept in probability theory. Two or more events are considered independent if the occurrence of one does not affect the occurrence of the other. In simpler terms, these are events where past outcomes do not influence future ones.

When we throw a pair of dice, what we get on one roll does not influence what will happen on the next roll - each is an independent event. This independence is critical to our binomial probability problem because it validates the assumption that the probability of rolling a seven remains constant (1/6) for each throw. Therefore, we’re able to use the same probability value for each calculation across all 120 trials in our exercise.