In the well known game of TattsLotto, the winning numbers are six number
selected without replacement from forty  five numbers, ie., 1; 2; : : : ; 45. Two
supplementary numbers are also selected without replacement. Each ticket
purchased consists of six numbers and it wins a money prize if these numbers
fall into any one of the following six divisions. A higher division will pay more
money than a lower division. The divisions are:
Division 1: six winning numbers;
Division 2:  five winning numbers plus one number from two supplementary
numbers;
Division 3: five winning numbers;
Division 4: four winning numbers;
Division 5: three winning numbers plus one or two supplementary numbers;
Division 6: one or two winning numbers plus two supplementary numbers.
(a) Find the probability of winning a prize in each division with the purchase
of a single ticket. -- ( will the remaining numbers be calculate dout of 39 or 37 in division 5?

To find the probability of winning a prize in each division with the purchase of a single ticket in TattsLotto, we need to calculate the number of ways the ticket numbers can match the winning numbers according to the rules for each division. Additionally, we must consider the total number of possible outcomes when choosing 6 numbers out of 45.

### Key Points:
- **Total number of combinations**: 
  \[
  \text{Total number of combinations} = \binom{45}{6}
  \]

### Division 1: All 6 Winning Numbers
- Probability of winning = 1 / \(\binom{45}{6}\)

### Division 2: 5 Winning Numbers + 1 Supplementary Number
- You must match 5 out of 6 winning numbers.
- The remaining number must be from the 2 supplementary numbers.
- Combinations:
  \[
  \binom{6}{5} \times \binom{39}{1} \times \binom{2}{1}
  \]
- The probability:
  \[
  P(\text{Division 2}) = \frac{\binom{6}{5} \times \binom{39}{1} \times \binom{2}{1}}{\binom{45}{6}}
  \]

### Division 3: 5 Winning Numbers
- You must match exactly 5 out of the 6 winning numbers.
- The remaining number cannot be from the 2 supplementary numbers.
- Combinations:
  \[
  \binom{6}{5} \times \binom{39}{1}
  \]
- The probability:
  \[
  P(\text{Division 3}) = \frac{\binom{6}{5} \times \binom{39}{1}}{\binom{45}{6}}
  \]

### Division 4: 4 Winning Numbers
- You must match exactly 4 out of the 6 winning numbers.
- The remaining 2 numbers can be any of the other 39 numbers.
- Combinations:
  \[
  \binom{6}{4} \times \binom{39}{2}
  \]
- The probability:
  \[
  P(\text{Division 4}) = \frac{\binom{6}{4} \times \binom{39}{2}}{\binom{45}{6}}
  \]

### Division 5: 3 Winning Numbers + 1 or 2 Supplementary Numbers
- You must match exactly 3 out of the 6 winning numbers.
- The remaining numbers include 1 or 2 supplementary numbers.
- There are two cases:
  - Case 1: 3 winning + 1 supplementary.
  - Case 2: 3 winning + 2 supplementary.
- Combinations:
  \[
  \binom{6}{3} \times \binom{39}{3} \times \left(\binom{2}{1} + \binom{2}{2}\right)
  \]
- The probability:
  \[
  P(\text{Division 5}) = \frac{\binom{6}{3} \times \left[\binom{39}{3} \times \left(\binom{2}{1} + \binom{2}{2}\right)\right]}{\binom{45}{6}}
  \]

### Division 6: 1 or 2 Winning Numbers + 2 Supplementary Numbers
- You must match exactly 1 or 2 out of the 6 winning numbers.
- The remaining numbers include both supplementary numbers.
- There are two cases:
  - Case 1: 1 winning number + 2 supplementary.
  - Case 2: 2 winning numbers + 2 supplementary.
- Combinations:
  \[
  \binom{6}{1} \times \binom{39}{5} \times \binom{2}{2} + \binom{6}{2} \times \binom{39}{4} \times \binom{2}{2}
  \]
- The probability:
  \[
  P(\text{Division 6}) = \frac{\binom{6}{1} \times \binom{39}{5} \times \binom{2}{2} + \binom{6}{2} \times \binom{39}{4} \times \binom{2}{2}}{\binom{45}{6}}
  \]

### Regarding the Calculation of Remaining Numbers
For Division 5, the calculation involves considering the remaining numbers from the total of 39 numbers, as the supplementary numbers have already been used.

Would you like to calculate the exact probabilities for each division?

Explore the probability of winning in each division of TattsLotto with a single ticket. Understand the combinatorial calculations required for Divisions 1 through 6, including exact formulas and probabilities. Perfect for understanding advanced combinatorial and probability concepts.

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