The Sorry State Lottery requires you to select 7 different numbers from 0 through 55. (Order is not important.) You are a Big Winner if the 7 numbers you select agree with those in the drawing, and you are a Small-Fry Winner if 6 of your 7 numbers agree with those in the drawing. (Round all answers to three significant figures. Enter the answers in scientific notation.)
What is the probability of being a Big Winner?
What is the probability of being a Small-Fry Winner?
What is the probability that you are either a Big Winner or a Small-Fry winner?
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To find the probabilities, we need to calculate the number of favorable outcomes and the total number of possible outcomes.
1. Probability of being a Big Winner:
To be a Big Winner, you must select all 7 numbers correctly. The order does not matter.
To calculate the number of favorable outcomes:
There is only one way to select all 7 numbers correctly.
To calculate the total number of possible outcomes:
You have to choose 7 numbers from a pool of 56 (0-55). The order does not matter.
We can use combinations to calculate this. The formula for combinations is nCr = n! / (r!(n-r)!)
n = 56 (total numbers to choose from)
r = 7 (numbers to be chosen)
nCr = 56! / (7!(56-7)!)
Calculating this, we get:
nCr = 56! / (7! × 49!) = 117,845,197
The probability of being a Big Winner is the number of favorable outcomes divided by the total number of possible outcomes:
Probability = 1 / 117,845,197 ≈ 8.48 × 10^(-9)
Therefore, the probability of being a Big Winner is approximately 8.48 × 10^(-9).
2. Probability of being a Small-Fry Winner:
To be a Small-Fry Winner, you must select 6 out of 7 numbers correctly. The order does not matter.
To calculate the number of favorable outcomes:
There are 7 numbers in total, and you need to select 6 of them correctly.
We can use combinations to calculate this: nCr = n! / (r!(n-r)!)
n = 7 (total numbers from which to choose)
r = 6 (numbers to be chosen)
nCr = 7! / (6!(7-6)!)
Calculating this, we get:
nCr = 7! / (6! × 1!) = 7
The probability of being a Small-Fry Winner is the number of favorable outcomes divided by the total number of possible outcomes:
Probability = 7 / 117,845,197 ≈ 5.94 × 10^(-8)
Therefore, the probability of being a Small-Fry Winner is approximately 5.94 × 10^(-8).
3. Probability of being either a Big Winner or a Small-Fry Winner:
Since being a Big Winner and a Small-Fry Winner are mutually exclusive events (you cannot be both at the same time), you can calculate the probability of being either by summing the two probabilities:
Probability = Probability of being a Big Winner + Probability of being a Small-Fry Winner
Probability = (1 / 117,845,197) + (7 / 117,845,197) ≈ 5.99 × 10^(-8)
Therefore, the probability of being either a Big Winner or a Small-Fry Winner is approximately 5.99 × 10^(-8).