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If π΄ and π΅ are two independent events from a sample space π such that the probability of π΅ is 0.6 and the probability of π΄ or π΅, or we could say probability of π΄ union π΅, is equal to 0.68, find the probability of π΄.
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Itβs important to know that π΄ and π΅ are two independent events.
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This means we can use specific formulas to help us find the probability of π΄.
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For two independent events, π΄ and π΅, the probability of π΄ or π΅ is equal to the probability of π΄ plus the probability of π΅ minus the probability of π΄ and π΅.
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And we also have the formula that the probability of π΄ and π΅ is equal to the probability of π΄ times the probability of π΅.
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So weβre asked to find the probability of π΄.
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Weβre given the probability of π΅.
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And weβre also given the probability of π΄ or π΅.
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And both of the given pieces of information are found in this first formula.
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So letβs use this one.
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So letβs begin replacing items.
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We know that the probability of π΄ or π΅ is 0.68.
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We are also given that the probability of π΅ is 0.6.
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We do not know the probability of π΄ and π΅.
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And weβre trying to find the probability of π΄.
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So we need to plug in something for this probability of π΄ and π΅.
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Well, we have this other formula.
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The probability of π΄ and π΅ can be replaced with the probability of π΄ times the probability of π΅.
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And we know the probability of π΅; itβs 0.6.
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Now, instead of writing probability of π΄ times 0.6, itβs more commonly written as 0.6 times the probability of π΄.
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Now, if we would rearrange these, where we have the probability of π΄ in the same spot but we switch the last two, so we have the negative 0.6 probability of π΄ and then plus 0.6.
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Notice both of these have a probability of π΄.
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So the first one that says probability of π΄ actually has a one in front.
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So we could take one times the probability of π΄ minus 0.6 times the probability of π΄.
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And one minus 0.6 would give us 0.4 times the probability of π΄.
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And then we bring down the 0.6.
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We also could have looked at this another way.
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We could have taken out what these two terms had in common, probability of π΄.
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And if we would take π of π΄, the probability of π΄, out, we would have one minus 0.6.
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And subtracting this, we get 0.4.
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So we will write this as 0.4 times the probability of π΄, just like we got.
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So now we can actually solve for the probability of π΄.
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Letβs begin by subtracting 0.6 from both sides of the equation.
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And doing so, we get 0.08 is equal to 0.4 times the probability of π΄.
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Divide both sides by 0.4, and we find that the probability of π΄ is 0.2.
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So, once again, the probability of π΄ is equal to 0.2.