# Out of 7 lottery tickets 3 are prize-winning tickets. If someone buys 4 tickets what is the probability of winning at least two prizes?

Let's separate the problem into four independent mutually exclusive cases:

Each of the above events has its own probability of occurrence. We are interested in events (c) and (d), the sum of the probabilities of their occurrence is what the problem is about. These two independent events constitute the event "winning at least two prizes". Since they are independent, a combined event's probability is a sum of its two components.

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To find the probability of winning at least two prizes out of 4 tickets, we can use the concept of complementary probability.

First, we find the probability of winning no prizes or only one prize, then subtract that from 1 to find the probability of winning at least two prizes.

The probability of winning no prizes out of 4 tickets can be calculated as: [ \frac{{\text{{Number of ways to choose 4 non-prize-winning tickets}}}}{{\text{{Total number of ways to choose 4 tickets}}}} ]

Similarly, the probability of winning exactly one prize out of 4 tickets can be calculated as: [ \frac{{\text{{Number of ways to choose 1 prize-winning ticket and 3 non-prize-winning tickets}}}}{{\text{{Total number of ways to choose 4 tickets}}}} ]

Once we have these probabilities, we subtract them from 1 to find the probability of winning at least two prizes.

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To find the probability of winning at least two prizes when buying 4 lottery tickets out of a total of 7, we can use the concept of combinations.

First, let's determine the total number of ways to choose 4 tickets out of 7, which is calculated using combinations:

[ \text{Total number of ways to choose 4 tickets out of 7} = \binom{7}{4} ]

Next, we need to consider the different scenarios where someone can win at least two prizes:

- Winning exactly 2 prizes out of the 3 prize-winning tickets and 2 non-prize-winning tickets.
- Winning exactly 3 prizes out of the 3 prize-winning tickets and 1 non-prize-winning ticket.
- Winning all 3 prizes out of the 3 prize-winning tickets and no non-prize-winning tickets.

We calculate the number of ways for each scenario using combinations, and then sum them up to find the total number of favorable outcomes.

Once we have the total number of favorable outcomes, we divide it by the total number of ways to choose 4 tickets out of 7 to find the probability.

Let's perform the calculations to find the probability of winning at least two prizes when buying 4 tickets.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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