WEBVTT
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An integer from one to 100 is chosen at random.
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Find the probability that it is part i) divisible by eight and part ii) not divisible by eight.
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We begin this question by recognizing that our sample space consists of the integers from one to 100.
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The elements in our sample space, which weβll call π, are therefore one, two, three, and so forth, all the way up to 100.
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And it should be easy to see that the number of elements in π is 100.
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Let us now define an event which weβll call π΄.
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We will say that π΄ is the event that the integer that we choose from our sample space is divisible by eight.
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To find the elements of set π΄, we now ask the question which of the integers from one to 100 will be divisible by eight.
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We now recall that, in order for a number to be divisible by eight with no remainder, by definition, it is part of the eight times table.
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We can then find the elements of set π΄ simply by writing out the eight times table.
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Eight times one is eight, eight times two is 16, eight times three is 24, and so on, all the way to eight times 12, which is 96.
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If we were to continue, we would get eight times 13, which is 104.
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Since 104 is greater than 100, it is not part of our sample space.
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So we get rid of this and stop at 96.
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We can now see that since we went from eight times one to eight times 12, we therefore have 12 elements in the set of π΄.
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And now on to probabilities, the question tells us that the integer from one to 100 is chosen at random.
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This means that each of the integers has an equal probability of being chosen.
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Because of this, we can find the probability of π΄ by dividing the number of elements of π΄ by the number of elements in our sample space.
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Subbing in the values that we have found, we find the probability of π΄ is therefore 12 over 100.
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And dividing the top and bottom half of this fraction by four, we get that the probability of π΄ is equal to three over 25.
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We therefore have our answer to part i of the question.
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The probability that the integer chosen is divisible by eight is three over 25.
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Letβs put this value to one side and now move on to part ii of the question.
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Find the probability that the integer chosen is not divisible by eight.
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Now weβve already defined that if event π΄ happens, the number chosen is divisible by eight.
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And therefore, if event π΄ does not happen, then our number will not be divisible by eight.
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To answer part b [part ii] of this question, we can therefore find the probability of the complement to event π΄, π΄ dash.
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Since the probabilities of all possible events sums to one, we know that the probability of π΄ dash is equal to one minus the probability of π΄.
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Weβve already found that the probability of event π΄ is three over 25.
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And we can sub this value in.
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A one is the same as 25 over 25.
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And therefore, our solution is 22 over 25.
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We have therefore answered part ii of the question.
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And we found that the probability that the integer chosen is not divisible by eight is 22 over 25.