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Instructor: Welcome back.
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In the beginning of the course
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we talked about a person's chances of winning the lottery
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but we didn't go through any actual calculations.
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For starters, let us define the rules of this lottery.
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We need to pick five numbers between 1 and 69
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and a Powerball number between 1 and 26.
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As we mentioned in the beginning of the course,
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the likelihood of two independent events
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occurring simultaneously
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equals the product of their individual probabilities.
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In this case one event would be
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guessing the Powerball number
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and the other would be getting the correct five numbers.
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Okay, let's start with the simpler part,
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picking our Powerball number.
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Since there is only one favorable outcome,
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the chance of picking the correct Powerball number
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is 1/26.
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Now, how about picking five numbers out of 69?
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As you know, order does not matter with lottery numbers
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so we would have to use combinations.
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Obviously, we cannot have the same value twice.
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This means that we are dealing with combinations
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without repetition.
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Therefore, let's apply the relevant formula.
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This suggests the total number of possible ways
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to pick our five numbers is 69 factorial
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over five factorial
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times 69 minus five,
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which is 64 factorial.
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The result is over 11 million.
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Therefore, we would've a lower chance than one in 11 million
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of correctly guessing the five numbers.
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To win the grand prize
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we would also have to correctly guess the Powerball number,
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so guessing all the numbers becomes 26 times less likely.
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Therefore there are almost 300 million
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different possible outcomes.
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Okay, to get the probability of winning
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we need two pieces of information
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number of favorable outcomes
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and the number of all possible outcomes.
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We already have the latter.
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What about the number of favorable outcomes?
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Well, they're going to be equal
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to the number of tickets we buy.
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Assuming we participate with a single ticket
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we can calculate the probability of winning.
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Using the favorable overall formula
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we get approximately one over 300 million.
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In other words, the probability of winning the lottery
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with a single ticket is approximately equal
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to a figure that is slightly above 0.000000003.
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Great.
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In this case, we had two events
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that needed to happen concurrently
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for us to win the big jackpot.
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The first one consisted of choosing the five numbers
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and the second one was picking the correct Powerball number.
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With this notion,
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we more or less exhausted the permutations,
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variations and combinations topic.
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In the next video, we're going to make a quick recap
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of everything we covered in this section
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and then we'll be ready to move forward.
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Thanks for watching.
5172
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