One test for ESP involves using Zener cards. Each card shows one of five different symbols
(square,circle,star,cross,wavy lines), and the person being tested has to predict the shape on each card before it is selected.
Find each of the probabilities requested for the person who has no ESP and is just guessing.
a) what is the probability of a correct prediction on any single trial?
b) what is the probability of correctly predicting exactly 20 cards in a series of n=100 trials?
c) what is the probability of correctly predicting more than 30 cards in a series of n=100 trials?
To find the probabilities for a person with no ESP who is just guessing, we need to first determine the probability of a correct guess on a single trial.
a) The probability of a correct prediction on any single trial is equal to the number of favorable outcomes divided by the total number of possible outcomes. Since there are 5 different symbols on the Zener cards, the probability of a correct guess is 1 out of 5:
Probability of a correct guess = 1/5 = 0.2 or 20%
b) To find the probability of correctly predicting exactly 20 cards in a series of 100 trials, we can use the binomial probability formula. The formula is given by:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of exactly k successes
- n is the total number of trials
- k is the number of successes
- p is the probability of success on a single trial
In this case, n = 100, k = 20, and p = 0.2. Substituting these values into the formula, we can calculate the probability:
P(X = 20) = (100C20) * (0.2)^20 * (1-0.2)^(100-20)
Note: (100C20) represents the binomial coefficient, also known as "n choose k," which calculates the number of ways to choose k successes out of n trials.
c) To find the probability of correctly predicting more than 30 cards in a series of 100 trials, we need to calculate the probability of correctly predicting 31, 32, 33,... up to 100 cards. We can then sum these individual probabilities.
Let's calculate the probability of correctly predicting exactly 31 cards and denote it as P(X > 30):
P(X > 30) = P(X = 31) + P(X = 32) + ... + P(X = 100)
Using the same binomial probability formula as before, we can substitute the values n = 100, k = 31, and p = 0.2 into the formula for each value of k from 31 to 100. Then, we sum all these probabilities to get the final result.
To calculate the probabilities requested, we need to understand the details of the experiment. In this case, the person being tested has no ESP and is randomly guessing the shape on each card. Let's break down each question and find the solutions.
a) Probability of a correct prediction on any single trial:
Since the person is randomly guessing, there are five possible outcomes, corresponding to the five shapes on the cards. The probability of correctly guessing the shape on any single trial is therefore 1 out of 5, or 1/5.
b) Probability of correctly predicting exactly 20 cards in a series of n=100 trials:
To calculate this probability, we need to consider the number of ways to choose 20 correct predictions out of 100 trials. Each trial has a 1/5 chance of being correct, so we can use the binomial probability formula.
The formula for the probability of getting exactly k successes in n independent trials with p probability of success on each trial is:
P(X=k) = (nCk) * p^k * (1-p)^(n-k)
In this case, n = 100, k = 20, and p = 1/5.
Using the binomial probability formula, we can calculate the probability of exactly 20 correct predictions in 100 trials.
P(X=20) = (100C20) * (1/5)^20 * (4/5)^80
c) Probability of correctly predicting more than 30 cards in a series of n=100 trials:
To calculate this probability, we need to sum up the probabilities of getting 31, 32, 33, ..., 100 correct predictions. Since each trial is independent, we can use the binomial probability formula to calculate each individual probability and add them up.
P(X>30) = P(X=31) + P(X=32) + ... + P(X=100)
In this case, n = 100 and p = 1/5. We need to calculate the probabilities for each k from 31 to 100 and add them up.
Now that we understand the setup and how to use the binomial probability formula, we can plug in the values and calculate the actual probabilities if needed.