# Probability – School lottery (basic)

This scenario tests basic arithmetic and probability calculation.
Difficulty level: Basic.
An advanced version of School lottery is also available.

Scenario:

Your school launches a spring lottery in which you can win various prizes. There are 900 lottery tickets, out of which 200 guarantee a prize. These include:

• single grand prize – a weekend trip,
• nine secondary prizes – free tickets to the movies,
• forty tertiary prizes – school T-shirts,
• the rest – school mugs.
1. What is the probability of winning each of the above prizes?
2. Say that you’ve arrived at the lottery late, and a third of all tickets have already been sold. But luck seems to be on your side – none of the prizes have been claimed! How has the probability of winning changed?
3. You’ve been dreaming of getting that school T-shirt for months now. By how much does your probability of winning increase if you buy two tickets at once instead of one, assuming a third of all tickets have still already been sold?

Useful calculators:

Question 1 hints:

Hint 1
How many winning tickets are there for each of the prizes? How many tickets are there in total?
Hint 2
The probability of an event is calculated by dividing the number of winning instances by the number of total possibilities.

Question 2 hints:

Hint 1
How many tickets are there in the pool now? How did the numbers of winning tickets change?
Hint 2
The probability of an event is calculated by dividing the number of winning instances by the number of total possibilities.

Question 3 hints:

Hint 1
How many pairs of tickets can you buy? How many of those are winning?
Hint 2
Observe that for every pair of tickets that give you a T-shirt, there are three possibilities. Not all of these instances give you a T-shirt, so you will have to work out the probability of each event.
Hint 3
If the first ticket grants you a T-shirt, the second one may or may not give you another T-shirt. And vice versa.

Solutions:

Question 1
P_trip = 1/900, P_movie = 1/100, P_Tshirt = 2/45, P_cup = 1/6
Question 2
P_trip = 1/600, P_movie = 3/200, P_Tshirt = 1/15, P_cup = 1/4
Question 3
P ≈ 0.13

Step-by-step solution:

Question 1
Let us begin by checking how many lottery tickets will give us a school mug. From the scenario, we know that there are in total 900 tickets, all = 900 tickets, and exactly 200 tickets that give a prize, winning = 200. Also, we know the prizes some of them give. In fact, we know the number of tickets granting all of the prizes, except the mugs. Therefore, find the number of mugs by subtracting the tickets we know from total number of winning tickets. In other words,
mugs = winning – trip – movie – Tshirt = 200 – 1 – 9 – 40 = 150.

We know that to get the probability that some event will happen, we need to divide the number of instances that support this event (the ones where the event does happen) by the number of all possible outcomes. In our case, the previous sentence translates to: the probability of winning a specific prize is the number of tickets that grant this prize divided by the number of total tickets in the pool.

Now that we know the number of tickets for each prize, we can use the probability calculator to compute:
P_trip = trip / all = 1/900,
P_movie = movie / all = 9 / 900 = 1/100,
P_Tshirt = Tshirt / all = 40 / 900 = 2/45,
P_mug = mug / all = 150 / 900 = 1/6.

Question 2
If a third of all tickets have already been sold, then two-thirds of the total number of tickets are still in the pool, which is
remaining = all * 2/3 = 900 * 2/3 = 600 tickets.

Note that among the 300 that have already been given away, none gave a prize. This means that there are still 200 prizes left to be won, in the same numbers as in Question 1. Therefore, if we want to calculate the new probabilities for each prize, then the number of chances for each prize has not changed, but the number of all possible outcomes (i.e., the number of all tickets in the pool) has changed from 900 to 600.

We are ready to use the probability calculator once again:
P_trip = trip / remaining = 1/600,
P_movie = movie / remaining = 9 / 600 = 3/200,
P_Tshirt = Tshirt / remaining = 40 / 600 = 1/15,
P_mug = mug / remaining = 150 / 600 = 1/4.

Question 3
If we buy two tickets at once and divide them into the first ticket to be drawn and the second ticket to be drawn, then we have four possibilities:
P₁. both tickets lose (no T-shirt for you),
P₂. the first ticket wins, the second loses,
P₃. the first ticket loses, the second wins,
P₄. both tickets win.
We want one of the last three to happen. Clearly, no two of them can happen at the same time. Also, since we want to win a T-shirt, we don’t mind which of the tickets grants us one, or even if both of them do. Therefore, the probability we want to find is
P = P₂ + P₃ + P₄.

Now we need to find P. First, however, note that we are looking for the probability of an event that happens when we buy two tickets. This means that the denominator (the number of all possible outcomes) is always the same. To be precise, all possible outcomes are all possible pairs of tickets from the pool of 600. Remember that we order the tickets we buy (we say which one is the first and the second), so we can use the combination calculator to see that there are exactly
pairs = 600 * 599
such pairs. If you’re having trouble understanding this, we can think that the first ticket is one of the 600 tickets available, and the second is one of the remaining 599.

Let’s move on to calculating P₂. In this case, the first (winning) ticket is one of the 40 that grant a T-shirt. The second (losing) ticket is one of the 599 – 39 = 560 tickets that do not give a T-shirt (it is not 600 – 40 = 560 as we just removed a T-shirt from the pool). Note here that we think of all the other prizes as losses since all we want is a T-shirt. Even if we win the main prize, we won’t be satisfied. That’s how unbelievably incredible these T-shirts are. Anyway, this means that we have 40 choices for the first ticket and 560 for the second one that fulfil our criteria, and therefore in total, there are 40 * 560 pairs that fit the description of option 2. The probability calculator gives:
P₂ = (40 * 560) / pairs = (40 * 560) / (600 * 599) ≈ 0.062.

We move on to P₃. Here we have a very similar situation, but this time we have 560 choices for the first (losing) ticket and 40 choices for the second (winning) ticket. Again using the probability calculator gives
P₃ = (560 * 40) / pairs = (560 * 40) / (600 * 599) ≈ 0.062.
Observe that we got the same as above. And it kind of makes sense – it doesn’t matter whether the first ticket wins and the second loses, or the other way round – there amount of pairs is the same.

Lastly, we consider P₄. In this case, we want the first ticket to be a winning one, so there are 40 choices for that to happen. The second ticket should also be winning, but now we have 39 choices for it because one of the 40 in the pool was the first ticket we bought. This gives
P₄ = (40 * 39) / pairs = (40 * 39) / (600 * 599) 0.004.

All in all, our probability of winning a T-shirt if we buy two tickets is
P = P₂ + P₃ + P₄ ≈ 0.062 + 0.062 + 0.004 = 0.13.

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