Some people believe in the force (yes, like in Star Wars). To test if young Anakin Skywalker has the force with him, he is told that some cards that Yoda can see but Anakin cannot contain a picture of a planet, a lightsaber, a laser rifle, or a spaceship. As Yoda looks at 20 such cards in turn, Anakin tries to guess what is on the card Yoda is looking at. Of course, Anakin has a 25% chance of simply guessing correctly.
a) Verify that the count of correct guesses in 20 cards follows a binomial distribution and write the notation.
b) What is the mean number of correct guesses?
c) What is the probability that Anakin guesses all 20 cards correctly?
d) Suppose Anakin guesses correctly on 10 of the cards. What is the probability of him doing this well or better by chance? Do you think he has the force with him?
a. x~bin(20,0.25)
b. µ = np = 20*0.25 = 5
c. P[20] = 0.25^20 = 9.0949e-13
d. P[≥10] = 0.0139 or 1.39%
so 10 guesses correct is unlikely by chance, but not that unlikely that i'd belirve that the force is with him !
a) To verify that the count of correct guesses in 20 cards follows a binomial distribution, we need to check if the following conditions are satisfied:
1. Each trial (guessing a card) is independent of each other.
2. There are only two possible outcomes in each trial: correct guess or incorrect guess.
3. The probability of success (correct guess) is constant for each trial.
4. The number of trials (20 cards) is fixed.
In this case, condition (1) is satisfied since each guess is independent of the others. Condition (2) is also satisfied since there are only two outcomes - correct guess or incorrect guess. Condition (3) is satisfied since the probability of a correct guess is fixed at 25% or 0.25. And finally, condition (4) is satisfied since the number of cards (trials) is fixed at 20.
Therefore, we can conclude that the count of correct guesses in 20 cards follows a binomial distribution.
The notation for this distribution can be written as X ~ B(20, 0.25), where X represents the number of correct guesses, 20 is the number of trials, and 0.25 is the probability of success (correct guess).
b) The mean number of correct guesses can be calculated using the formula:
Mean (μ) = Number of trials (n) * Probability of success (p)
In this case, the number of trials is 20 and the probability of success (correct guess) is 0.25.
Mean (μ) = 20 * 0.25 = 5
Therefore, the mean number of correct guesses is 5.
c) To calculate the probability that Anakin guesses all 20 cards correctly, we can use the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
In this case, k (the number of correct guesses) is 20, n (the number of trials) is 20, and p (the probability of success/correct guess) is 0.25.
P(X = 20) = (20 choose 20) * 0.25^20 * (1 - 0.25)^(20 - 20)
Using the combination formula and calculating the probabilities, we find:
P(X = 20) ≈ 0.0000009537
Therefore, the probability that Anakin guesses all 20 cards correctly is approximately 0.0000009537.
d) To find the probability of him doing as well or better by chance, we need to calculate the cumulative probability of guessing 10 or more correct cards. This can be done by summing up the individual probabilities for 10, 11, 12,..., 20 correct guesses.
P(X ≥ 10) = P(X = 10) + P(X = 11) + P(X = 12) + ... + P(X = 20)
Using the binomial probability formula and calculating the probabilities for each case, we find:
P(X ≥ 10) ≈ 0.050831
Therefore, the probability of Anakin guessing 10 or more cards correctly by chance is approximately 0.050831.
Based on this probability, it seems unlikely that Anakin has the force with him, as the probability is relatively low (less than 0.05). However, it's important to note that this is just a statistical analysis and does not determine whether Anakin actually has the force or not.
a) To determine if the count of correct guesses in 20 cards follows a binomial distribution, we need to verify the following conditions:
1) Each trial (guess) is independent. The outcome of each card guess does not affect the outcome of any other card guess.
2) There are only two possible outcomes for each trial: a correct guess or an incorrect guess.
3) The probability of success (correct guess) remains the same for each trial.
In this case, Anakin has a 25% chance of guessing correctly, which is the same for each trial. Therefore, the count of correct guesses in 20 cards follows a binomial distribution.
Notation: X ~ B(20, 0.25)
b) The mean number of correct guesses can be calculated using the formula:
Mean (μ) = n * p
Where n is the number of trials (20 cards) and p is the probability of success (25%).
Mean (μ) = 20 * 0.25 = 5
Therefore, the mean number of correct guesses is 5.
c) The probability that Anakin guesses all 20 cards correctly can be calculated using the binomial probability formula:
P(X = k) = C(n, k) * p^k * q^(n-k)
Where P(X = k) is the probability of getting exactly k successes in n trials, C(n, k) is the combination function, p is the probability of success for each trial, and q is the probability of failure (1 - p).
In this case, we need to calculate P(X = 20):
P(X = 20) = C(20, 20) * (0.25^20) * (0.75^(20-20))
P(X = 20) = 1 * (0.25^20) * (0.75^0)
P(X = 20) = 0.25^20
The probability that Anakin guesses all 20 cards correctly is 0.25^20, which is extremely low.
d) To find the probability of Anakin guessing correctly on 10 or more cards by chance, we need to calculate the cumulative probability:
P(X >= 10) = P(X = 10) + P(X = 11) + ... + P(X = 20)
Using the binomial probability formula, we can calculate each term and sum them up.
P(X >= 10) = ∑[i=10 to 20] C(20, i) * (0.25^i) * (0.75^(20-i))
Calculating this sum will give us the probability of Anakin doing as well as or better than guessing 10 cards correctly by chance alone.
Based on the calculated probability, we can assess whether Anakin has the Force with him.