Question

Seventy percent of the bicycles sold by a certain store are mountain bikes. Among 100 randomly selected bike purchases, what is the approximate probability that a. At most 75 are mountain bikes? (Hint: See Example 6.33 ) b. Between 60 and 75 (inclusive) are mountain bikes? c. More than 80 are mountain bikes? d. At most 30 are not mountain bikes?

Short answer

The calculations will yield the following results: a. The probability that at most 75 are mountain bikes b. The probability that between 60 and 75 (inclusive) are mountain bikes c. The probability that more than 80 are mountain bikes d. The probability that at most 30 are not mountain bikes Please note that the exact numerical results are omitted here as they depend on the calculations made.

Step-by-step solution

Calculation of Probability for 'At Most 75 Mountain Bikes'

Applying the binomial probability formula for each amount of bikes from 0 to 75, and then adding them all together to get the cumulative probability. Let's call the outcome 'a'.

Calculation of Probability for 'Between 60 and 75 (inclusive) Mountain Bikes'

Similarly apply the binomial probability formula inside this range and then add all the results together. Call this outcome 'b'.

Calculation of Probability for 'More Than 80 Mountain Bikes'

Here we apply the binomial probability formula from 81 to 100, because 'more than 80' means starting from 81. Again, we add them all together. Let's call this outcome 'c'.

Calculation of Probability for 'At Most 30 Bikes are not Mountain Bikes'

This is equivalent to 'at least 70 Mountain Bikes', thus apply the binomial probability formula from 70 to 100 and then add them together. Call this result 'd'.

Key concepts

Probability Distribution

A probability distribution is a mathematical description that shows all the possible outcomes of a random event and how likely each outcome is to occur. It tells us how probabilities are distributed over the values of the random variable. Think of it as a list that details, for each possible outcome, the chance of that outcome happening.

For example, if we flip a fair coin, there are two possible outcomes: heads or tails. Each has a probability of 0.5, or 50%. In this simple distribution, the probabilities sum to 1, meaning one of the outcomes is certain to occur. For more complex scenarios, like the number of mountain bikes sold, probability distributions can help us understand the expected patterns in the randomness of the sales.

Cumulative Probability

Cumulative probability is the sum of the probabilities of all outcomes up to and including a certain value. It's like adding up the probabilities of each individual outcome, one after the other, until you reach that value. This helps to answer questions about the likelihood of a variable falling within a certain range.

When calculating cumulative probabilities for a range of outcomes, we can either add up the probabilities of the individual outcomes manually, or, when dealing with a binomial distribution, we might use statistical tables or software to simplify the process. For instance, if a question asks about the probability of selling at most 75 mountain bikes, you use cumulative probability to sum up the likelihoods of selling 0 to 75 bikes.

Binomial Distribution

The binomial distribution represents the probabilities of the outcomes for a fixed number of independent trials, each with two possible outcomes—commonly called success and failure. When you hear binomial, think 'bi-' for two, like the two outcomes. Important features are the number of trials (n), the probability of success in a single trial (p), and the probability of failure (q, which can be calculated as 1 - p).

The probability of getting exactly 'k' successes in 'n' trials is given by the binomial probability formula: \[ P(X = k) = {\binom{n}{k}} p^k (1-p)^{n-k} \]
Utilities of binomial distribution extend to many fields, including quality control, survey sampling, and here, for predicting the number of mountain bikes sold. This distribution helps us to calculate probabilities like the ones in our exercise—giving us the chance of selling a certain number of mountain bikes out of a set number of sales.