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# Question Video: Determining the Probability of Union of Two Events Mathematics

π΄ and π΅ are two events in a sample space of a random experiment where π(π΄) = 3/10, π(π΅) = 1/5, and π(π΄ β π΅) = 1/10. Find π(π΄ βͺ π΅).

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### Video Transcript

π΄ and π΅ are two events in a sample space of a random experiment, where the probability of π΄ is three-tenths, the probability of π΅ is one-fifth, and the probability of π΄ minus π΅ is one-tenth. Find the probability of π΄ union π΅.

In this question, we are asked to find the probability of π΄ union π΅. This is the probability that event π΄ or event π΅ or both occur. And the addition rule of probability states that the probability of π΄ union π΅ is equal to the probability of π΄ plus the probability of π΅ minus the probability of π΄ intersection π΅. In this question, we are given values for the probability of π΄ and the probability of π΅. However, we are not told the probability of π΄ intersection π΅, where this is the probability that both π΄ and π΅ occur.

We are told that the probability of π΄ minus π΅ is one-tenth. This is the probability that π΄ occurs but π΅ does not as shown on the Venn diagram. This can be calculated by subtracting the probability of π΄ intersection π΅ from the probability of π΄. Substituting the values we know into this equation, we have one-tenth is equal to three-tenths minus the probability of π΄ intersection π΅. This can be rearranged so that the probability of π΄ intersection π΅ is equal to three-tenths minus one-tenth, which is equal to two-tenths or one-fifth.

We now have the three values that enable us to calculate the probability of π΄ union π΅. This is equal to three-tenths plus one-fifth minus one-fifth. As the one-fifths cancel, we are just left with three-tenths or 0.3. If the probability of π΄ is three-tenths, the probability of π΅ is one-fifth, and the probability of π΄ minus π΅ is one-tenth, then the probability of π΄ union π΅ is three-tenths.

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