Mary has a bag with 8 tickets : 3 are red and 5 are blue. She picks 5 tickets, one at a time, and places them back in the bag each time. She counts the number of red tickets she picks.
Does this situation fit the binomial model ?
A random variable X is called a binomial random variable if it meets the following conditions:
- There are a fixed number of trials, n, of a random phenomenon.
- There are only two possible outcomes on each trial, often called success and failure.
- The probability of success, p, is constant for each trial.
- Each trial is independent of the other trials.
In our problem:
- Mary picks 5 tickets, thus n=5.
- There are only 2 possible outcomes on each trial: pick a red ticket or pick a blue ticket.
- In this situation, red is considered success and blue is considered failure. Because Mary puts the ticket back after each trial, the probability of success (picking a red ticket) is constant for each trial: p=\dfrac{3}{8}.
- Because Mary puts the ticket back after each trial, the trials are independent.
This situation fits the binomial model with n=5 and p=\dfrac{3}{8}.
We roll a die 9 times and count how many times we get 6.
Does this situation fit the binomial model ?
A random variable X is called a binomial random variable if it meets the following conditions:
- There are a fixed number of trials, n, of a random phenomenon.
- There are only two possible outcomes on each trial, often called success and failure.
- The probability of success, p, is constant for each trial.
- Each trial is independent of the other trials.
In our problem:
- We roll the die 9 times, thus n=9.
- There are only 2 possible outcomes on each trial: rolling a 6 or rolling any other number.
- In this situation, rolling a 6 is considered a success and rolling anything else is considered a failure.
- p=\dfrac{1}{6}.
- The trials are independent.
This situation fits the binomial model, with n=9 and p=\dfrac{1}{6}.
In a standard card deck, we pull out ten cards one by one without replacing them and count the number of aces.
Does this situation fit the binomial model ?
A random variable X is called a binomial random variable if it meets the following conditions:
- There are a fixed number of trials, n, of a random phenomenon.
- There are only two possible outcomes on each trial, often called success and failure.
- The probability of success, p, is constant for each trial.
- Each trial is independent of the other trials.
In our problem, since we do not replace the cards, the probability of success (drawing an ace) changes each time. The situation is not a binomial model.
The situation is not a binomial model.
The probability that an archer hits a target is 0.7. He shoots 15 times and counts the number of times that he hits the target.
Does this situation fit the binomial model ?
A random variable X is called a binomial random variable if it meets the following conditions:
- There are a fixed number of trials, n, of a random phenomenon.
- There are only two possible outcomes on each trial, often called success and failure.
- The probability of success, p, is constant for each trial.
- Each trial is independent of the other trials.
In our problem:
- He shoots 15 times, thus n=15.
- There are only 2 possible outcomes on each trial: hitting the target or not.
- Hitting the target is considered a success and the probability of success is constant for each trial: p=0.7.
- The trials are independent.
This situation fits the binomial model, with n=15 and p=0.7.
We flip two coins simultaneously 10 times and count the number of times that two heads come up.
Does this situation fit the binomial model ?
A random variable X is called a binomial random variable if it meets the following conditions:
- There are a fixed number of trials, n, of a random phenomenon.
- There are only two possible outcomes on each trial, often called success and failure.
- The probability of success, p, is constant for each trial.
- Each trial is independent of the other trials.
In our problem:
- We flip the coin ten times, thus n=10.
- There are only 2 possible outcomes on each trial: two heads is success and any other result is failure.
- The probability of success is constant for each trial: p=\dfrac{1}{4}.
- The trials are independent.
This situation fits the binomial model, with n=10 and p=\dfrac{1}{4}.
In a penalty shoot-out in a soccer match, each team takes turns shooting 5 penalty kicks. Assume that the probability that our goal keeper saves a penalty is 0.2.
Does this situation fit the binomial model ?
A random variable X is called a binomial random variable if it meets the following conditions:
- There are a fixed number of trials, n, of a random phenomenon.
- There are only two possible outcomes on each trial, often called success and failure.
- The probability of success, p, is constant for each trial.
- Each trial is independent of the other trials.
In our problem:
- The number of kicks is 5, thus n=5.
- There are only 2 possible outcomes on each trial: saving the penalty kick is success and failing to save is failure.
- The probability of success (saving the penalty kick) is: p=0.2.
- The trials are independent.
This situation fits the binomial model, with n=5 and p=0.2.
We roll a die 20 times. We count the number of times that the number we get is greater than the one in the previous roll (we don't count the first roll).
Does this situation fit the binomial model ?
A random variable X is called a binomial random variable if it meets the following conditions:
- There are a fixed number of trials, n, of a random phenomenon.
- There are only two possible outcomes on each trial, often called success and failure.
- The probability of success, p, is constant for each trial.
- Each trial is independent of the other trials.
In our problem the trials are not independent, since the trial i affects the trial i+1.
This situation does not fits the binomial model.