in a test for esp a subject is told that cards the experimenter can see but he cannot contain either a star, a circle, a wave, or a square. as the experimenter looks at each of 20 cards in turn, the subject names the shape on the card. a subject who is just guessing has probability .25 of guessing correctly on each card.
a)what is the mean number of correct guesses in many repititions of the experiment?
b)what is the probability of exactly 5 correct guesses?
To answer these questions, let's break them down step by step.
a) What is the mean number of correct guesses in many repetitions of the experiment?
To determine the mean number of correct guesses, we need to multiply the probability of guessing a card correctly by the total number of cards.
In this case, the probability of guessing a card correctly is 0.25. And the total number of cards is 20.
So, the mean number of correct guesses can be calculated as follows:
Mean number of correct guesses = Probability of guessing correctly * Total number of cards
Mean number of correct guesses = 0.25 * 20 = 5
Therefore, the mean number of correct guesses in many repetitions of the experiment is 5.
b) What is the probability of exactly 5 correct guesses?
To find the probability of exactly 5 correct guesses, we will use the binomial probability formula.
The binomial probability formula is given by:
P(x) = (nCx) * p^x * (1 - p)^(n - x)
Where:
- P(x) is the probability of getting exactly x correct guesses.
- n is the total number of trials (number of cards in this case).
- x is the number of successful outcomes (number of correct guesses in this case).
- p is the probability of a single successful outcome (probability of guessing correctly in this case).
- (nCx) represents the binomial coefficient, which is the number of ways to choose x successes from n trials.
Substituting the values into the formula, we get:
P(5) = (20C5) * (0.25^5) * (1 - 0.25)^(20 - 5)
Calculating the binomial coefficient and evaluating the formula, we can determine the probability:
P(5) = 0.0141 (approximately)
Therefore, the probability of exactly 5 correct guesses is approximately 0.0141.
To solve this problem, let's first define some variables:
Let X be the random variable representing the number of correct guesses.
Let p be the probability of guessing correctly on each card, which is 0.25.
a) To find the mean number of correct guesses, we need to find the expected value E(X).
The expected value E(X) is equal to the sum of all possible values of X multiplied by their respective probabilities. In this case, the possible values of X range from 0 to 20.
E(X) = 0 * P(X = 0) + 1 * P(X = 1) + 2 * P(X = 2) + ... + 20 * P(X = 20)
Now, the probability P(X = x) can be calculated using the binomial probability formula:
P(X = x) = (nCx) * (p^x) * ((1-p)^(n-x))
Where n is the total number of cards (20) and x is the number of correct guesses.
Using this formula, we can find the mean number of correct guesses.
E(X) = 0 * P(X = 0) + 1 * P(X = 1) + 2 * P(X = 2) + ... + 20 * P(X = 20)
E(X) = Σ [x * P(X = x)] (from x = 0 to 20)
Calculating this sum will give us the mean number of correct guesses.
b) To find the probability of exactly 5 correct guesses, we can use the same binomial probability formula:
P(X = 5) = (20C5) * (p^5) * ((1-p)^(20-5))
We can substitute the values of n, x, and p into the formula to calculate the probability.
mean: .25*20
exactly five in a row: (.25)^5