The Baltimore Ravens play 17 football games per season. Let’s define X to be the number of games in which they win the pregame coin toss in a full season. (For those of you who do not follow football, the pregame coin toss is used to determine which team receives the ball first and on which end of the field each of the two teams start.)
Does the random variable X have a binomial distribution?
Yes, the random variable X has a binomial distribution.
Yes, the random variable X, which represents the number of games in which the Baltimore Ravens win the pregame coin toss in a full season, can have a binomial distribution.
For a random variable to have a binomial distribution, it must satisfy the following conditions:
1. Each trial must have two possible outcomes: success (winning the coin toss) or failure (losing the coin toss).
2. The probability of success (p) must remain constant for each trial.
3. The trials must be independent of each other (the outcome of one trial should not affect the outcome of another trial).
4. The variable of interest is the number of successes (X) in a fixed number of trials.
In the case of the Baltimore Ravens, each game can be considered a trial with two possible outcomes: winning or losing the coin toss. The probability of winning the coin toss remains constant for each game, and the outcomes of one game do not influence the outcomes of other games. Therefore, the number of games in which the Ravens win the pregame coin toss follows a binomial distribution.
To determine if the random variable X has a binomial distribution, we need to check if it satisfies the following criteria:
1. The experiments are identical: In this case, each coin toss before a game is considered an identical experiment since the circumstances and chances are the same for each toss.
2. The experiments have two possible outcomes: The coin toss can only result in two outcomes - either the Ravens win the toss or they lose the toss.
3. The probability of success is constant: The probability of winning the coin toss is the same for every game since the circumstances and rules remain constant throughout the season.
4. The trials are independent: The outcome of one coin toss does not affect the outcome of the others. Each game is treated as an independent event.
Based on these criteria, the random variable X, which represents the number of games in which the Baltimore Ravens win the pregame coin toss, does have a binomial distribution.