Many people play the lottery in hopes that their financial troubles will be over. However, the lottery is very unlikely to help change a player’s financial situation. In many regions, lottery companies have to report on the back of a scratch ticket the proportion of players who win, but
The basic idea behind the calculation of a probability is to divide an amount you’re interested in by the total number of possibilities. If a simple lottery game lets you choose three out of ten numbers and only one number is a winner, then the probability of winning is
\[\frac{3}{10} = 0.3\]
or 30%.
We can do some simple calculations to help us determine whether a lottery game is worth playing. The expected value of a lottery game is the amount of money we expect to win or lose if we play the game many times.
Let’s say the above game costs $5 to play. If you win, you get your $5 back plus $7 more. If you lose, you lose your $5 that you paid to play. The expected value is found by multiplying the dollar amount of each possible outcome by the probability of that outcome, then adding these products up. For this game, the expected value is
\[-\$5 \cdot 0.7 + \$7 \cdot 0.3\]
One outcome of the game is that we would lose. That means we would lose $5, hence, the negative sign. We multiply this outcome by the probability of losing, \(0.7\). Another outcomes is that they win the game. They would gain $7. That outcome has a probability of 0.3.
\[-\$5 \cdot 0.7 + \$7 \cdot 0.3 = -1.4\]
The game has an expected value of \(-\$1.40\). That means we should expect to lose, on average, $1.40 per game. Put in another way, if we played this game 100 times, we expect to lose \(100 \cdot \$1.40 = \$140\).
1.
A person rolls a six sided die. They need a 5 or 6 in order to win
the game. What’s the probability that they win?
To calculate a probability of winning (when every option has an equal chance of happening), divide the number of ways of winning by the number of total options. There are 2 out of 6 numbers that cause us to win.
\[\frac{2}{6} \approx 0.3333\]
2.
A player has to roll a one in a board game in order for their
character to make it to the winning location. Their previous rolls have
been a 2, 3, 2, 5, 4, 5, 3, 4, 6, and 6. What’s the probability that the
next roll is a 1?
This is a bit of a trick question, as the previous rolls do not matter. The probability of getting a 1 is still 1 out of 6, or 0.16667 approximately. Human nature thinks we are bound to win at some point if we have lost repeatedly before, but the outcome of one dice roll does not impact the chances of a certain outcome of another dice roll.
3.
The back of a particular \$\(2\)
scratch ticket reveals that there is a \(0.1\) probability of making your money
back, a \(0.2\) probability of winning
your money back plus $2, and a \(0.05\)
chance of winning your money back plus $50. The only other outcome of
the game is to lose the $2 that a player started with. What’s the
expected value of this game?
To calculate an expected value, multiply each value by its probability, then add the results.
There’s a \(0.1\) probability of walking away with $0 (winning and getting our money back). There’s a \(0.2\) probability of walking away with \(2\). There’s a \(0.05\) probability of walking away with $50. Losing is the only other option. We subtract all the probabilities from 1 (which represents 100%) to calculate the probability of losing.
\[1 - 0.1 - 0.2 - 0.05 = 0.65\]
There’s a \(0.65\) probability of losing the $2.
We multiply each of these values by their probability, then add the results.
\[\$0 \cdot 0.1 + \$2 \cdot 0.2 + \$50 \cdot 0.05 + -\$2 \cdot 0.65 =\]
\[\$0 + \$0.40 + \$2.50 - \$1.3 = \$1.60\]
This game has a positive expected value of $1.60. That means we should play this game as we
4.
In the game described in part 3, how much money should you expect
to win or lose if you bought 100 tickets?
The expected value for one game is $1.60. If we play this game 100 times, the expected winnings are
\[\$1.60 \cdot 100 = \$160\]
1. A person rolls a six sided die. They need a 3, 4, 5, or 6 in order to win the game. What’s the probability that they win?
2. The back of a particular \(2\) scratch ticket reveals that there is a \(0.1\) probability of making your money back, a \(0.1\) probability of winning your money back plus $2, and a \(0.025\) chance of winning your money back plus $30. The only other outcome of the game is to lose the $2 that a player started with. What’s the expected value of this game?
3. In the game described in part 3, how much money should you expect to win or lose if you bought 100 tickets?
1. Can you find any examples where people "gamed" the system and played a lottery or gambling game with a positive expected value?