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In Tattslotto, a player picks a selection of six numbers from the numbers 1 to 4...
Jan 29, 2024
In Tattslotto, a player picks a selection of six numbers from the numbers 1 to 45. To determine the winners, eight numbers are chosen at random – the first six are designated as the winning numbers, and the other two as the supplementary numbers. Prizes are determined as follows. Division 1: 6 winning numbers Division 2: 5 winning numbers and 1 supplementary Division 3: 5 winning numbers Division 4: 4 winning numbers Division 5: 3 winning numbers and 1 supplementary Find the number of combinations which satisfy each of the divisions, and hence the probabilities of winning each of the five divisions.
Solution by Steps
step 1
To find the number of combinations for Division 1, we use the binomial coefficient for choosing 6 winning numbers out of 45
step 2
The binomial coefficient is calculated as nCr(45,6) nCr(45, 6)
step 3
Using the asksia-ll calculator, nCr(45,6)=8145060 nCr(45, 6) = 8145060
step 4
The probability of winning Division 1 is the reciprocal of the number of combinations, 1nCr(45,6) \frac{1}{nCr(45, 6)}
step 5
Using the asksia-ll calculator, the probability is 1.2277380399898834385504833604663440171097573252990155996395...×10−7 1.2277380399898834385504833604663440171097573252990155996395... \times 10^{-7}
Answer
The number of combinations for Division 1 is 8145060, and the probability of winning is approximately 1.227738×10−7 1.227738 \times 10^{-7} .
Key Concept
Calculating combinations and probabilities for lottery winnings using binomial coefficients.
Explanation
The number of combinations for Division 1 is found using the binomial coefficient formula, and the probability is the reciprocal of this number.
step 1
For Division 2, we need to calculate the combinations for 5 winning numbers and 1 supplementary number
step 2
First, we calculate the combinations for choosing 5 out of 45 numbers and then multiply by the combinations for choosing 1 out of the remaining 2 supplementary numbers
step 3
Using the asksia-ll calculator, nCr(45,5)×nCr(2,1)=2443518 nCr(45, 5) \times nCr(2, 1) = 2443518
step 4
The probability of winning Division 2 is the reciprocal of the number of combinations, 1nCr(45,5)×nCr(2,1) \frac{1}{nCr(45, 5) \times nCr(2, 1)}
step 5
Using the asksia-ll calculator, the probability is 4.0924601332996114618349445348878133903658577509967186654651...×10−7 4.0924601332996114618349445348878133903658577509967186654651... \times 10^{-7}
Answer
The number of combinations for Division 2 is 2443518, and the probability of winning is approximately 4.092460×10−7 4.092460 \times 10^{-7} .
Key Concept
Calculating combinations and probabilities for lottery winnings with supplementary numbers.
Explanation
For Division 2, we calculate the combinations for the main and supplementary numbers separately and then find the probability.
step 1
For Division 3, we calculate the combinations for choosing 5 winning numbers out of 45
step 2
Using the asksia-ll calculator, nCr(45,5)=1221759 nCr(45, 5) = 1221759
step 3
The probability of winning Division 3 is the reciprocal of the number of combinations, 1nCr(45,5) \frac{1}{nCr(45, 5)}
step 4
Using the asksia-ll calculator, the probability is 8.1849202665992229236698890697756267807317155019934373309302...×10−7 8.1849202665992229236698890697756267807317155019934373309302... \times 10^{-7}
Answer
The number of combinations for Division 3 is 1221759, and the probability of winning is approximately 8.184920×10−7 8.184920 \times 10^{-7} .
Key Concept
Calculating combinations and probabilities for lottery winnings without supplementary numbers.
Explanation
For Division 3, we only consider the combinations for the main numbers to find the probability.
step 1
For Division 4, we calculate the combinations for choosing 4 winning numbers out of 45
step 2
Using the asksia-ll calculator, nCr(45,4)=148995 nCr(45, 4) = 148995
step 3
The probability of winning Division 4 is the reciprocal of the number of combinations, 1nCr(45,4) \frac{1}{nCr(45, 4)}
step 4
Using the asksia-ll calculator, the probability is 6.7116346186113627974093090372160139602000067116346186113627...×10−6 6.7116346186113627974093090372160139602000067116346186113627... \times 10^{-6}
Answer
The number of combinations for Division 4 is 148995, and the probability of winning is approximately 6.711635×10−6 6.711635 \times 10^{-6} .
Key Concept
Calculating combinations and probabilities for a smaller set of winning numbers in a lottery.
Explanation
For Division 4, we find the combinations for 4 winning numbers and then calculate the probability.
step 1
For Division 5, we calculate the combinations for choosing 3 winning numbers and 1 supplementary number
step 2
First, we calculate the combinations for choosing 3 out of 45 numbers and then multiply by the combinations for choosing 1 out of the remaining 2 supplementary numbers
step 3
Using the asksia-ll calculator, nCr(45,3)×nCr(2,1)=28380 nCr(45, 3) \times nCr(2, 1) = 28380
step 4
The probability of winning Division 5 is the reciprocal of the number of combinations, 1nCr(45,3)×nCr(2,1) \frac{1}{nCr(45, 3) \times nCr(2, 1)}
step 5
Using the asksia-ll calculator, the probability is 0.0000352360817477096546863988724453840732910500352360817477096546... 0.0000352360817477096546863988724453840732910500352360817477096546...
Answer
The number of combinations for Division 5 is 28380, and the probability of winning is approximately 0.0000352361 0.0000352361 .
Key Concept
Calculating combinations and probabilities for lottery winnings with fewer numbers and a supplementary number.
Explanation
For Division 5, we calculate the combinations for the main and supplementary numbers separately and then find the probability.
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