How to find probability distributions for discrete random variables - AP Statistics

There are four conditions that need to be satisfied for a binomial experiment:

1. Each trial must have two outcomes.

2. Each trial must be independent.

3. All trials must be identical.

4. The probabilities of the outcomes remain constant must not change with each trial.

The only choice that satisfies all four of these conditions (and is therefore a binomial experiment) is the rock-paper-scissors scenario.

To calculate Prob(at least one bull's eye), we can instead compute one minus the complementary probability, P(no bull's eye).

So we have P(at least one bull's eye)=1-P(no bull's eye).

The chance of getting no bull's eyes is .

This means the probability of getting at least one bull's eye is

The movements that this particle can make include: RRL, RLR, LRR.

The chance of getting RRL is . This is also the chance of getting any of those movements.

To get the total probability, we can add up the individual probabilities since the events are all mutually exclusive.

Thus, we get the following as the solution.

To calculate this probability, we need to calculate the chance of getting 4 tails and then a head.

Each tail has a prob. of and a head is , so we multiply to the power of 4 (because we need 4 tails) by (for the single head).

So the probability is

.

A discrete variable is a variable which can only take a countable number of values. For example, the number of times that a coin can come up heads in ten flips can only be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10. Thus, there are a countable number of possible outcomes (in this case 11). This is true for coin flips, but not for caterpillar length, water flow, or rates of return for stocks.

Two steps are crucial here.

First, we need to recognize this is a binomial distribution with and .

Second, we need to realize we can use a normal approximation of the binomial since and , which are both larger than 5.

With that said, we can calculate a -score and its -value, keeping in mind that our mean will be and our standard deviation will be , which is about .

There are four conditions that need to be satisfied for a binomial experiment:

1. Each trial must have two outcomes.

2. Each trial must be independent.

3. All trials must be identical.

4. The probabilities of the outcomes remain constant must not change with each trial.

The only choice that satisfies all four of these conditions (and is therefore a binomial experiment) is the rock-paper-scissors scenario.

To calculate Prob(at least one bull's eye), we can instead compute one minus the complementary probability, P(no bull's eye).

So we have P(at least one bull's eye)=1-P(no bull's eye).

The chance of getting no bull's eyes is .

This means the probability of getting at least one bull's eye is

The movements that this particle can make include: RRL, RLR, LRR.

The chance of getting RRL is . This is also the chance of getting any of those movements.

To get the total probability, we can add up the individual probabilities since the events are all mutually exclusive.

Thus, we get the following as the solution.

To calculate this probability, we need to calculate the chance of getting 4 tails and then a head.

Each tail has a prob. of and a head is , so we multiply to the power of 4 (because we need 4 tails) by (for the single head).

So the probability is

.

A discrete variable is a variable which can only take a countable number of values. For example, the number of times that a coin can come up heads in ten flips can only be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10. Thus, there are a countable number of possible outcomes (in this case 11). This is true for coin flips, but not for caterpillar length, water flow, or rates of return for stocks.

Two steps are crucial here.

First, we need to recognize this is a binomial distribution with and .

Second, we need to realize we can use a normal approximation of the binomial since and , which are both larger than 5.

With that said, we can calculate a -score and its -value, keeping in mind that our mean will be and our standard deviation will be , which is about .

There are four conditions that need to be satisfied for a binomial experiment:

1. Each trial must have two outcomes.

2. Each trial must be independent.

3. All trials must be identical.

4. The probabilities of the outcomes remain constant must not change with each trial.

The only choice that satisfies all four of these conditions (and is therefore a binomial experiment) is the rock-paper-scissors scenario.

To calculate Prob(at least one bull's eye), we can instead compute one minus the complementary probability, P(no bull's eye).

So we have P(at least one bull's eye)=1-P(no bull's eye).

The chance of getting no bull's eyes is .

This means the probability of getting at least one bull's eye is

The movements that this particle can make include: RRL, RLR, LRR.

The chance of getting RRL is . This is also the chance of getting any of those movements.

To get the total probability, we can add up the individual probabilities since the events are all mutually exclusive.

Thus, we get the following as the solution.

To calculate this probability, we need to calculate the chance of getting 4 tails and then a head.

Each tail has a prob. of and a head is , so we multiply to the power of 4 (because we need 4 tails) by (for the single head).

So the probability is

.

A discrete variable is a variable which can only take a countable number of values. For example, the number of times that a coin can come up heads in ten flips can only be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10. Thus, there are a countable number of possible outcomes (in this case 11). This is true for coin flips, but not for caterpillar length, water flow, or rates of return for stocks.

Two steps are crucial here.

First, we need to recognize this is a binomial distribution with and .

Second, we need to realize we can use a normal approximation of the binomial since and , which are both larger than 5.

With that said, we can calculate a -score and its -value, keeping in mind that our mean will be and our standard deviation will be , which is about .

There are four conditions that need to be satisfied for a binomial experiment:

1. Each trial must have two outcomes.

2. Each trial must be independent.

3. All trials must be identical.

4. The probabilities of the outcomes remain constant must not change with each trial.

The only choice that satisfies all four of these conditions (and is therefore a binomial experiment) is the rock-paper-scissors scenario.