If one event, say Black, has P(.70) and another event, say Red, has P(.30). What is the probability in ten events of alternating choices, Red then Black, and being 100% correct, 90% correct. What should be the expected % correct and what is the standard deviation?
To calculate the probability of getting a certain number of correct alternations in ten events, we can use the binomial probability formula. The formula for calculating the probability of exactly k successes in n independent Bernoulli trials is:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of getting exactly k successes.
- n is the total number of trials.
- k is the number of successful outcomes.
- p is the probability of success in a single trial.
In this case, we will consider getting a Red event as a success and getting a Black event as a failure.
Let's calculate the probabilities for each scenario:
1. 100% Correct:
To have 100% correct, we need exactly 5 Red and 5 Black events in any order. Therefore, k=5.
P(X = 5) = (10C5) * (0.3)^5 * (0.7)^(10-5)
2. 90% Correct:
To have 90% correct, we can have 9 correct alternations and 1 incorrect alternation, where the incorrect alternation can happen in any of the 10 events. Therefore, k=9.
P(X = 9) = (10C9) * (0.3)^9 * (0.7)^(10-9)
To calculate the expected percentage of correct alternations and the standard deviation, we need to consider the probabilities for all possible outcomes.
Expected Percentage of Correct Alternations:
The expected percentage of correct alternations is the sum of the products of each number of correct alternations and its corresponding probability, multiplied by 100.
Expected % Correct = (5 * P(X = 5) + 9 * P(X = 9)) * 100
Standard Deviation:
To calculate the standard deviation, we need to find the variance first, which is the sum of the squares of the differences between each outcome and the expected percentage of correct alternations, multiplied by their probabilities.
Variance = ( (5 - Expected % Correct)^2 * P(X = 5) + (9 - Expected % Correct)^2 * P(X = 9) )
Finally, the standard deviation is the square root of the variance.
Standard Deviation = sqrt(Variance)
Using these calculations, you can find the expected percentage of correct alternations and the standard deviation for these scenarios.