A fast food chain claims that all of its quick win soda cups are winners. (There is a little tab you pull off the side of the cup which says either you are a winner or not. Lisa bought 8 sodas to try her luck.
What is the probability that at least 5cups were winners?
What is the probability that between 3 and 6 of the cups 9inclusive) were winners.
What is the probability that at most three cups were winners?
Is reasonable to approximate the distribution using a normal curve? Why or why not?
To solve these probability questions, we need to use the concept of binomial distribution. In this case, we have a random experiment (buying sodas) with two possible outcomes (winning or not winning), and each outcome has a fixed probability (all cups are winners).
The probability of success, denoted as p, is the probability of winning a soda cup. Since the chain claims that all cups are winners, p = 1.
Now let's answer each question step by step:
1. Probability that at least 5 cups were winners:
To find the probability of at least 5 cups being winners, we need to calculate the probability of exactly 5, exactly 6, exactly 7, and exactly 8 cups being winners, and then sum up these probabilities.
P(at least 5 cups are winners) = P(exactly 5) + P(exactly 6) + P(exactly 7) + P(exactly 8)
To calculate the probability of exactly "k" successes in "n" trials (in this case, k winners in 8 trials), we use the formula:
P(exactly k) = C(n, k) * p^k * (1-p)^(n-k)
Where C(n, k) represents the number of combinations of n items taken k at a time, given by the formula:
C(n, k) = n! / (k! * (n-k)!)
Using this formula, we can calculate each probability and then sum them up.
2. Probability that between 3 and 6 (inclusive) cups were winners:
Similar to the first question, we need to calculate the probabilities of exactly 3, 4, 5, and 6 cups being winners, and then sum up these probabilities.
P(between 3 and 6 cups are winners) = P(exactly 3) + P(exactly 4) + P(exactly 5) + P(exactly 6)
Again, we use the binomial distribution formula to calculate each probability.
3. Probability that at most three cups were winners:
To find the probability of at most 3 cups being winners, we need to calculate the probability of exactly 0, exactly 1, exactly 2, and exactly 3 cups being winners, and then sum up these probabilities.
P(at most 3 cups are winners) = P(exactly 0) + P(exactly 1) + P(exactly 2) + P(exactly 3)
Once again, we use the binomial distribution formula to calculate each probability.
Now, regarding the approximation of the distribution using a normal curve, we can use the normal approximation to the binomial distribution when both np and n(1-p) are greater than or equal to 5. In this case, n is 8 and p is 1, so both np and n(1-p) are greater than 5. Therefore, it is reasonable to approximate the distribution using a normal curve.