Question

Austin is baking 2 batches of cookies. since he tends to be quite forgetful, there's a good chance he might burn the cookies, and then they won't come out tasty. each batch is independent, and the probability of his first batch being tasty is 30%, and the probability of his second batch being tasty is 80%. if x is a random variable that represents the number of tasty batches of cookies alex makes, what are the probabilities for all possible values of x.

Answer 1

Austin's probabilities for making tasty batches of cookies are 14% for making 0 tasty batches, 62% for making 1 tasty batch, and 24% for making 2 tasty batches.

Step-by-step explanation:

To calculate the probability distribution for the random variable X, which represents the number of tasty batches of cookies Austin makes, we need to consider all possible outcomes. Austin can make 0, 1, or 2 tasty batches, with X taking on these values. We'll calculate the probabilities as follows:

• P(X = 0): The probability that neither batch is tasty. This is the product of the probabilities that each individual batch is not tasty.
• P(X = 1): The probability that exactly one batch is tasty. There are two scenarios for this: either the first batch is tasty and the second is not, or the second batch is tasty and the first is not.
• P(X = 2): The probability that both batches are tasty. This is the product of the individual probabilities that each batch is tasty.

Let's calculate these probabilities:

• 
  P(X = 0) = (1 - 0.30) × (1 - 0.80) = 0.70 × 0.20 = 0.14 or 14%
• 
  P(X = 1) = (0.30 × (1 - 0.80)) + ((1 - 0.30) × 0.80) = (0.30 × 0.20) + (0.70 × 0.80)
  = 0.06 + 0.56 = 0.62 or 62%
• 
  P(X = 2) = 0.30 × 0.80 = 0.24 or 24%

Here we have the probability distribution:

• P(X = 0) = 14%
• P(X = 1) = 62%
• P(X = 2) = 24%