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I heard many people say that it is better to play 100 tickets in one draw of 5/49 lottery, rather than 100 plays/draws with 1 ticket.

Is there any scientific basis for this, or actually there is no difference?

triplodrindl
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1 Answers1

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It sort of depends, however, it is usually better to spread out your tickets over the 100 lotteries, rather than place all of your tickets into one...

Let's say that you have $n$ lotteries $L_1 , L_2, \dots, L_n$ to enter, where in your case, $n=100$. Let $n_i$ be the total number of tickets in the $i^{th}$ lottery (before you add your tickets). Let $t_{i}$ be the number of tickets you enter into the $i^{th}$ lottery.

The probability that you win the $i^{th}$ lottery is $$p_{i} = \frac{t_{i}}{n_{i}+t_{i}}$$

where the $t_{i}$ in the denomenator comes from the fact that you are adding $t_i$ tickets to the lottery.

The expected number of lotteries that you win is $$\mathbb{E}[Lotteries\ I \ Win]=\sum_{i=1}^{n} p_i$$

Now, suppose that $n_{i} = 50$ for all $i$. That is, each lottery has exaclty 50 tickets entered into it. Then, if you place all of your tickets into lottery 1, the probability you win is $p_1 = 100/150 = 2/3$ and $p_j =0 $ for all $j\geq2$. Thus, the expected number of lotteries you win is $2/3$. However, if you place exactly one ticket into each of the 100 lotteries, then the expected number of lotteries you win is $1/51 \cdot n = 100/51 \approx 2$, so in this case, you were better off placing one ticket in each lottery than you were placing all of your tickets into one. In general, your strategy will depend on the expected number of tickets entered into each lottery. Intuitively, you probably want to place a few more tickets into lotteries where few people are playing, and hold back on those that have a lot of participants. If you assume that the same number of tickets $N$ have been entered into each lottery, then you actually maximize your expected winnings by placing one ticket into each lottery, and minimize your chances of winning by placing all of your tickets into exactly one (i.e. if the lotteries are identical, then the worst thing you can do is place all of your tickets into one). To see this, note that $$\frac{100}{N+100}\leq \frac{100}{k} \cdot \frac{k}{N+k}$$ for any $1\leq k \leq 100$. Sort of seems unintuitive at first, but the math doesn't lie :).

mm8511
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