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Suppose 𝐴, 𝐵, and 𝐶 are three mutually exclusive events in a sample space 𝑆.
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Given that 𝑆 is equal to 𝐴 union 𝐵 union 𝐶, the probability of 𝐴 is one-fifth the probability of 𝐵, and the probability of 𝐶 is equal to four multiplied by the probability of 𝐴, find the probability of 𝐵 union 𝐶.
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We are told in this question that our three events are mutually exclusive.
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We recall that two or more events are mutually exclusive if they cannot happen at the same time.
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This can be represented on a Venn diagram as shown, where there is no overlap or intersection between the three circles which represent the events 𝐴, 𝐵, and 𝐶.
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Since the sample space is equal to 𝐴 union 𝐵 union 𝐶, then the probability of 𝐴 union 𝐵 union 𝐶 is one.
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The sum of the probabilities inside our circles must equal one.
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We also recall that if we have two mutually exclusive events 𝑋 and 𝑌, then the probability of 𝑋 union 𝑌 is equal to the probability of 𝑋 plus the probability of 𝑌.
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This means that in our question, the probability of 𝐴 plus the probability of 𝐵 plus the probability of 𝐶 must equal one.
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We are told that the probability of 𝐴 is equal to one-fifth of the probability of 𝐵.
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If we multiply both sides of this equation by five, we have the probability of 𝐵 is equal to five multiplied by the probability of 𝐴.
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We can substitute this into our equation.
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We can also replace the probability of 𝐶 by four multiplied by the probability of 𝐴.
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Our equation becomes the probability of 𝐴 plus five multiplied by the probability of 𝐴 plus four multiplied by the probability of 𝐴 is equal to one.
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The left-hand side simplifies to 10 multiplied by the probability of 𝐴.
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We can then divide through by 10 so that the probability of 𝐴 is equal to one-tenth.
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This could also be written as the decimal 0.1.
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The probability of 𝐵 is equal to five multiplied by this.
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This is equal to five-tenths, which simplifies to one-half.
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The probability of event 𝐵 is one-half.
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In the same way, the probability of event 𝐶 is four-tenths.
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Dividing the numerator and denominator by two, this is equal to two-fifths.
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We have been asked to calculate the probability of 𝐵 union 𝐶.
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Using our rule for mutually exclusive events, this is equal to the probability of 𝐵 plus the probability of 𝐶.
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Using the two fractions with a common denominator, we have five-tenths plus four-tenths, which is equal to nine-tenths.
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The probability of 𝐵 union 𝐶, where the two events are mutually exclusive, is nine-tenths.
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This is the sum of the probabilities inside the two circles in our Venn diagram.