What's the difference between sample space and event in a lottery game (where 50 numbers are drawn and you have to guess/pick six numbers?). Can you give examples of both?
Suppose that an experiment has five separate mutually exclusive outcomes: A, B, C, D, and E. If the sample space for the experiment is a uniform sample space, what is P (A or E)?
In a lottery game where 50 numbers are drawn and players have to pick six numbers, the sample space refers to the entire set of possible outcomes. It contains every combination of six numbers that could be drawn. On the other hand, an event is a specific outcome or set of outcomes that we are interested in.
Let's take an example to understand this further:
Sample Space: The sample space in this case is the set of all possible combinations of six numbers that can be drawn from the 50 available numbers. It would contain combinations like (1, 2, 3, 4, 5, 6), (7, 8, 9, 10, 11, 12), (13, 14, 15, 16, 17, 18), and so on, up to (45, 46, 47, 48, 49, 50). The total number of elements in the sample space is determined by the formula for combinations: C(50, 6) = 15,890,700.
Event: An event is a specific outcome or set of outcomes within the sample space that we are interested in. In this lottery game example, let's say we are interested in the event of guessing all six numbers correctly. So, the event in this case would be winning the jackpot, which is picking the exact six numbers that get drawn. The probability of winning the jackpot is typically very low due to the large size of the sample space.
Another example of an event could be guessing at least three numbers correctly. This event would encompass all combinations where the player guesses three, four, five, or six numbers correctly. The probability of this event happening is generally higher than that of winning the jackpot.
To summarize, the sample space in a lottery game with 50 numbers and six picks consists of all possible combinations of six numbers. An event is a specific outcome or set of outcomes within the sample space that we are interested in, such as winning the jackpot or guessing a certain number of numbers correctly.