Bernoulli trial

Answer: A Bernoulli trial, named after Swiss mathematician Jacob Bernoulli, is a random experiment with two possible outcomes: success or failure. It is a fundamental concept in probability theory and statistics.

The knowledge points that Bernoulli trial contains are:

1. Random experiment: A Bernoulli trial is a random experiment, meaning that the outcome is uncertain and cannot be predicted with certainty.

2. Two possible outcomes: In a Bernoulli trial, there are only two possible outcomes - success or failure. These outcomes are mutually exclusive, meaning that only one of them can occur at a time.

3. Independent trials: Each Bernoulli trial is independent of the others, meaning that the outcome of one trial does not affect the outcome of any other trial.

The formula for Bernoulli trial is as follows:

P(X = k) = (n choose k) * p^k * (1-p)^(n-k)

Where:

• P(X = k) is the probability of getting exactly k successes in n trials.
• (n choose k) is the binomial coefficient, which represents the number of ways to choose k successes from n trials.
• p is the probability of success in a single trial.
• (1-p) is the probability of failure in a single trial.
• k is the number of successes.

To apply the Bernoulli trial formula, you need to know the values of n, p, and k. Substitute these values into the formula to calculate the probability of getting exactly k successes in n trials.

The symbol for Bernoulli trial is X, which represents the random variable that counts the number of successes in a given number of trials.

There are several methods for Bernoulli trial, including:

1. Using the formula: As mentioned earlier, you can use the formula to calculate the probability of getting a specific number of successes in a given number of trials.

2. Using a probability table: If the number of trials is small, you can create a probability table to calculate the probabilities of different outcomes.

3. Using a calculator or software: There are various calculators and software programs available that can perform the calculations for you.

Here are two solved examples on Bernoulli trial:

Example 1: A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads?

Solution: In this case, n = 5 (number of trials), p = 0.5 (probability of success - getting a head), and k = 3 (number of successes). Plugging these values into the formula, we get:

P(X = 3) = (5 choose 3) * 0.5^3 * (1-0.5)^(5-3) = 10 * 0.125 * 0.25 = 0.3125

Therefore, the probability of getting exactly 3 heads in 5 coin tosses is 0.3125.

Example 2: A basketball player has a free throw success rate of 80%. If she takes 10 free throws, what is the probability of making at least 8 of them?

Solution: In this case, n = 10 (number of trials), p = 0.8 (probability of success - making a free throw), and k = 8, 9, 10 (number of successes). We need to calculate the probabilities of making 8, 9, and 10 free throws and then add them together.

P(X >= 8) = P(X = 8) + P(X = 9) + P(X = 10) = (10 choose 8) * 0.8^8 * (1-0.8)^(10-8) + (10 choose 9) * 0.8^9 * (1-0.8)^(10-9) + (10 choose 10) * 0.8^10 * (1-0.8)^(10-10) = 0.302 + 0.268 + 0.107 = 0.677

Therefore, the probability of making at least 8 out of 10 free throws is 0.677.

Now, let's move on to some practice problems on Bernoulli trial:

1. A fair six-sided die is rolled 4 times. What is the probability of getting exactly 2 sixes?

2. A bag contains 10 red balls and 5 blue balls. Two balls are drawn at random without replacement. What is the probability of getting exactly 1 red ball?

3. A multiple-choice test has 10 questions, each with 4 possible answers. If a student guesses the answers to all the questions, what is the probability of getting at least 8 correct answers?

FAQ on Bernoulli trial:

Q: What is the difference between a Bernoulli trial and a binomial distribution? A: A Bernoulli trial refers to a single experiment with two possible outcomes, while a binomial distribution refers to the probability distribution of the number of successes in a fixed number of independent Bernoulli trials.

Q: Can a Bernoulli trial have more than two outcomes? A: No, a Bernoulli trial can only have two outcomes - success or failure.

Q: Can the probability of success in a Bernoulli trial change? A: Yes, the probability of success can vary from trial to trial, but it remains constant within a single trial.

Q: Can a Bernoulli trial be dependent on previous trials? A: No, a Bernoulli trial is assumed to be independent of any previous or future trials.

Q: Can a Bernoulli trial have a negative outcome? A: No, a Bernoulli trial can only have two outcomes - success or failure. Negative outcomes are not considered in this context.