Final answer:
The probability that Atreyu selects each possible number of singing birthday cards from a local greeting card store can be found using the binomial probability formula.
Step-by-step explanation:
The probability of Atreyu selecting a singing card on any given year is 43%. In the 10 years since he began this tradition, we need to find the probability of selecting each possible number of singing birthday cards, denoted by x. To find these probabilities, we can use the binomial probability formula:
P(x) = C(n, x) * p^x * (1-p)^(n-x)
Where n is the total number of trials (10 in this case), p is the probability of success on each trial (0.43), and C(n, x) is the number of ways to choose x successes out of n trials.
Let's calculate the probabilities for each value of x:
- P(x = 0) = C(10, 0) * 0.43^0 * (1-0.43)^(10-0) = 0.0007
- P(x = 1) = C(10, 1) * 0.43^1 * (1-0.43)^(10-1) = 0.0067
- P(x = 2) = C(10, 2) * 0.43^2 * (1-0.43)^(10-2) = 0.0333
- P(x = 3) = C(10, 3) * 0.43^3 * (1-0.43)^(10-3) = 0.1047
- P(x = 4) = C(10, 4) * 0.43^4 * (1-0.43)^(10-4) = 0.2133
- P(x = 5) = C(10, 5) * 0.43^5 * (1-0.43)^(10-5) = 0.2835
- P(x = 6) = C(10, 6) * 0.43^6 * (1-0.43)^(10-6) = 0.2417
- P(x = 7) = C(10, 7) * 0.43^7 * (1-0.43)^(10-7) = 0.1344
- P(x = 8) = C(10, 8) * 0.43^8 * (1-0.43)^(10-8) = 0.0404
- P(x = 9) = C(10, 9) * 0.43^9 * (1-0.43)^(10-9) = 0.0050
- P(x = 10) = C(10, 10) * 0.43^10 * (1-0.43)^(10-10) = 0.0002