Oz Lotto

Last night for $1.30, I bought a single game in Oz Lotto. The winning numbers were 1, 4, 13, 19, 26, 31, 35 and the numbers in bold were the matching numbers that I had in my game. I got three out of the seven numbers which didn't win me a prize but it was impressive nonetheless I thought. What were the odds of this happening?

The number of ways that seven numbers can be selected from 45 numbers, disregarding order, is given by \( ^{45}C_7  = 45,379,620 \). Thus there is only one chance in 45,379,620 that the combination 1, 4, 13, 19, 26, 31, 35 will appear. Given those seven numbers, in how many ways can three of them be chosen? Clearly, in \( ^7C_3 \) = 35 ways.

This means that the number of ways in which three winning numbers can occur out of seven numbers is  \( \text{35 x }  ^{38}C_4 \). This is because none of the remaining four numbers can contain a winning number, thus it is a choice of any four of the remaining 38 non-winning numbers. The probability of choosing three winning numbers in a single game is thus: \[ \text{35 x } \frac{^{38}C_4}{^{45}C_7} = \frac{172235}{3025308} \approx 0.05693 \]Thus the figure as a percentage is about 5.7%. I dithered around trying to work this out but I'm fairly certain that this is the correct approach. For the sake of completeness, I'll work out the probability of getting four, five and six numbers:

For four numbers, chances are: \[ \text{35 x } \frac{^{38}C_3}{^{45}C_7} = \frac{4921}{756327} \approx 0.0065 \]For five numbers, chances are: \[ \text{21 x } \frac{^{38}C_2}{^{45}C_7} = \frac{4921}{15126540} \approx 0.0003 \]For six numbers, chances are:\[ \text{7 x } \frac{^{38}C_1}{^{45}C_7} = \frac{133}{22689810} \text{  or a very small chance!} \]Working on this post, I noticed that Blogger's rendering of LaTeX code has changed. Before, use of the $ sign at the beginning and end of a mathematical expression meant that MathJax was used to create the inline expression:

MathJax is an open-source JavaScript display engine for LaTeX, MathML, and AsciiMath notation that works in all modern browsers ... MathJax uses web-based fonts (in those browsers that support it) to produce high-quality typesetting that scales and prints at full resolution (unlike mathematics included as images) source

Now however, use of the dollar sign creates an inline image of the mathematical expression and it is only by use of the curved brackets and backslash that MathJax is invoked. The double dollar sign to create display expressions remains unchanged however, and there is no need yet to use backslash and open and closed square brackets. As before, in Android, nothing seems to work.