The probability that an individual egg in a carton of eggs is cracked is 0.05. You have picked
out a carton of 1 dozen eggs (that’s 12 eggs) at the grocery store. Determine the probability
that at most two of the eggs in the carton are cracked
To determine the probability that at most two of the eggs in the carton are cracked, we need to calculate the probability of three scenarios:
1. Zero eggs cracked
2. One egg cracked
3. Two eggs cracked
Let's calculate each scenario step by step:
1. Probability of zero eggs cracked:
The probability that an individual egg is cracked is 0.05. Therefore, the probability that an individual egg is not cracked is 1 - 0.05 = 0.95.
To calculate the probability that all 12 eggs are not cracked, we multiply the individual probabilities of each egg not being cracked:
Probability of zero eggs cracked = (0.95)^12
2. Probability of one egg cracked:
We need to calculate the probability of exactly one egg being cracked and the remaining 11 eggs not being cracked.
We can choose 1 egg out of 12 in 12C1 = 12 ways. The individual probability of that egg being cracked is 0.05, and the probability of the remaining 11 eggs not being cracked is 0.95.
Probability of one egg cracked = 12C1 * (0.05)^1 * (0.95)^11
3. Probability of two eggs cracked:
Similarly, we need to calculate the probability of exactly two eggs being cracked and the remaining 10 eggs not being cracked.
We can choose 2 eggs out of 12 in 12C2 = 66 ways. The individual probability of each egg being cracked is 0.05, and the probability of the remaining 10 eggs not being cracked is 0.95.
Probability of two eggs cracked = 12C2 * (0.05)^2 * (0.95)^10
Now, add up the probabilities from the three scenarios to find the probability that at most two eggs in the carton are cracked:
Probability of at most two eggs cracked = Probability of zero eggs cracked + Probability of one egg cracked + Probability of two eggs cracked
I hope this explanation helps!
To determine the probability that at most two eggs in the carton are cracked, we need to calculate the sum of the probabilities of different scenarios: zero cracked eggs, one cracked egg, and two cracked eggs.
Let's start by calculating the probability of zero cracked eggs:
The probability of an individual egg being cracked is 0.05 or 5%. Therefore, the probability of an individual egg not being cracked is 1 - 0.05 = 0.95 or 95%. Since there are 12 eggs in the carton, the probability of all 12 eggs not being cracked is 0.95^12.
Next, let's calculate the probability of exactly one cracked egg:
We can select any one egg out of the 12 eggs to be cracked, and the probability of that egg being cracked is 0.05. The remaining 11 eggs are not cracked, so the probability of exactly one cracked egg is 12 * 0.05 * 0.95^11.
Finally, let's calculate the probability of exactly two cracked eggs:
We can select any two eggs out of the 12 eggs to be cracked. The probability of both these selected eggs being cracked is (0.05^2). The remaining 10 eggs are not cracked, so the probability of exactly two cracked eggs is (12C2) * (0.05^2) * (0.95^10), where (12C2) represents the number of ways to choose 2 objects out of 12 without considering their order.
To find the probability that at most two eggs in the carton are cracked, we need to add up these three probabilities:
Probability of at most two cracked eggs = probability of zero cracked eggs + probability of exactly one cracked egg + probability of exactly two cracked eggs
Probability of at most two cracked eggs = 0.95^12 + 12 * 0.05 * 0.95^11 + (12C2) * (0.05^2) * (0.95^10)
Now, you can calculate the probability using the above formula and determine the result.