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

𝐴 and 𝐡 are two events in a sample space of a random experiment where 𝑃(𝐴) = 3/10, 𝑃(𝐡) = 1/5, and 𝑃(𝐴 βˆ’ 𝐡) = 1/10. Find 𝑃(𝐴 βˆͺ 𝐡).

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Video Transcript

𝐴 and 𝐡 are two events in a sample space of a random experiment, where the probability of 𝐴 is three-tenths, the probability of 𝐡 is one-fifth, and the probability of 𝐴 minus 𝐡 is one-tenth. Find the probability of 𝐴 union 𝐡.

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

We are told that the probability of 𝐴 minus 𝐡 is one-tenth. This is the probability that 𝐴 occurs but 𝐡 does not as shown on the Venn diagram. This can be calculated by subtracting the probability of 𝐴 intersection 𝐡 from the probability of 𝐴. Substituting the values we know into this equation, we have one-tenth is equal to three-tenths minus the probability of 𝐴 intersection 𝐡. This can be rearranged so that the probability of 𝐴 intersection 𝐡 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 𝐴 union 𝐡. 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 𝐴 is three-tenths, the probability of 𝐡 is one-fifth, and the probability of 𝐴 minus 𝐡 is one-tenth, then the probability of 𝐴 union 𝐡 is three-tenths.

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