Question

The National Hockey League (NHL) has about \(70 \%\) of its players born outside the United States, and of those born outside the United States, approximately \(60 \%\) were born in Canada. \({ }^{5}\) Suppose that \(n=12\) NHL players are selected at random. Let \(x\) be the number of players in the sample born outside of the United States so that \(p=.7,\) and find the following probabilities: a. At least five or more of the sampled players were born outside the United States b. Exactly seven of the players were born outside the United States c. Fewer than six were born outside the United States.

Short answer

Question: In a basketball team of 12 players, 70% of the players were born outside the United States. Calculate the probability of the following scenarios: a) At least five or more players were born outside the United States, b) Exactly seven players were born outside the United States, and c) Fewer than six players were born outside the United States. Answer: a) For at least five or more players born outside the U.S., the probability is given by \(P(X\geq5) = \sum_{x=5}^{12} \binom{12}{x} (0.7)^x (0.3)^{12-x}\). b) For exactly seven players born outside the U.S., the probability is given by \(P(X=7) = \binom{12}{7} (0.7)^7 (0.3)^{12-7}\). c) For fewer than six players born outside the U.S., the probability is given by \(P(X<6) = \sum_{x=0}^{5} \binom{12}{x} (0.7)^x (0.3)^{12-x}\).

Step-by-step solution

a. At least five or more of the sampled players were born outside the United States

We want to find the probability that five or more players (out of 12) are born outside the U.S. Since this condition includes all the scenarios from \(x=5\) to \(x=12\), we need to calculate the sum of probabilities for each value of \(x\) in this range. Therefore, the required probability is: \(P(X\geq5) = P(X=5) + P(X=6) + \cdots + P(X=12)\). We will calculate each probability using the binomial formula and add them: \(P(X\geq5) = \sum_{x=5}^{12} \binom{12}{x} (0.7)^x (0.3)^{12-x}.\)

b. Exactly seven of the players were born outside the United States

We want to know the probability of exactly seven players being born outside the U.S. We will use the binomial distribution formula for \(x=7\): \(P(X=7) = \binom{12}{7} (0.7)^7 (0.3)^{12-7}.\)

c. Fewer than six were born outside the United States

We want to find the probability that fewer than six players were born outside the U.S. Since this condition includes all the scenarios from \(x=0\) to \(x=5\), we need to calculate the sum of probabilities for each value of \(x\) in this range. Therefore, the required probability is: \(P(X<6) = P(X=0) + P(X=1) + \cdots + P(X=5)\). We will calculate each probability using the binomial formula and add them: \(P(X<6) = \sum_{x=0}^{5} \binom{12}{x} (0.7)^x (0.3)^{12-x}.\)

Key concepts

Binomial Distribution Formula

The binomial distribution is a fundamental concept in statistics, used to model the number of successes in a fixed number of independent trials with the same probability of success. For instance, if you flip a coin multiple times, each flip represents an independent trial with two possible outcomes: heads or tails.

The formula for calculating the probability of exactly 'k' successes in 'n' trials is given by:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
Where:\

• \( \binom{n}{k} \) is the binomial coefficient, representing the number of ways to choose 'k' successes from 'n' trials.
• \( p \) is the probability of success on a single trial.
• \( 1-p \) is the probability of failure on a single trial.

Application to the NHL Example

In the NHL example, the exercise is to calculate probabilities related to the number of players born outside the United States among a randomly selected sample of 12 players. The success in this scenario is a player being born outside the U.S., with a probability of 0.7 (since 70% are foreign-born). Each player selected can be seen as a 'trial,' and the total number of players we're interested in (being foreign-born) would be the 'successes'.

The probabilities are then computed using the binomial formula for different values of 'k', which in this case refers to the number of foreign-born players in our sample.

Probability of Events

The probability of an event is a measure of the likelihood that the event will occur. Probabilities are expressed on a scale from 0 to 1, where 0 means the event will not occur, and 1 indicates certainty that the event will occur. If all outcomes are equally likely, the probability of an event happening is the number of desired outcomes divided by the total number of possible outcomes.

When using the binomial distribution formula, we're calculating the probability for the number of successes in a set number of trials to be a specific value, like finding the chances of getting exactly seven players born outside the U.S. out of twelve picked at random.

Combining Probabilities

Problems often ask for the probability of a range of outcomes, such as getting 'at least' or 'fewer than' a certain number of successes. This requires adding up the probabilities for all relevant outcomes. For our 'at least five' NHL scenario, we would calculate and sum probabilities from five players up to all twelve. Conversely, for 'fewer than six', we'd sum up from zero to five players.

Random Sampling

Random sampling is a statistical method used to select a subset of individuals from a larger population, wherein each individual has an equal chance of being chosen. This ensures that the sample represents the population fairly, allowing for generalizations about the population based on the sample.

In the context of the NHL example, random sampling refers to selecting the 12 NHL players without bias or preference, so that each player (regardless of nationality) has an equal chance of being included in the sample. The concept assumes that each player's selection is independent, meaning one player’s nationality does not influence another’s chance of selection.

Importance in Probability Calculation

Random sampling is essential for applying the binomial distribution formula accurately. If the sampling were not random, the formula's assumptions about independence and probability wouldn't hold true, leading to incorrect probability calculations. The approach to random sampling, therefore, underpins the validity of statistical inferences made from the sample data.