Video Transcript
A small choir has a tenor singer, three soprano singers, a baritone singer, and a mezzo-soprano singer. If one of their names was chosen randomly, determine the probability that it was the name of the tenor singer or soprano singer.
In this question, we are told that the choir has one tenor singer. It has three soprano singers, one baritone singer, and one mezzo-soprano singer. This means that there are a total of six singers in the choir. We need to find the probability that a randomly chosen singer was either a tenor or soprano singer. We can do this directly from the question. As there was one tenor singer and three soprano singers, this is a total of four singers.
We know that the probability of an event occurring can be written as a fraction, where the numerator is the number of successful outcomes and the denominator is the number of possible outcomes. The probability of selecting a tenor or soprano singer is therefore equal to four out of six, or four-sixths. As both the numerator and denominator are divisible by two, this simplifies to two-thirds. The probability that the selected singer was a tenor singer or soprano singer is two-thirds.
An alternative method here is to use the addition rule of probability, which states that the probability of 𝐴 union 𝐵 is equal to the probability of 𝐴 plus the probability of 𝐵 minus the probability of 𝐴 intersection 𝐵. We will let 𝐴 be the event of selecting a tenor singer and 𝐵 be the event of selecting a soprano singer. The probability of event 𝐴 is therefore equal to one-sixth, and the probability of event 𝐵 is equal to three-sixths. Whilst this fraction simplifies to one-half, we will leave the denominators the same at this stage.
The events 𝐴 and 𝐵 are mutually exclusive as they cannot happen at the same time. There is no singer that is a tenor singer and a soprano singer. We know that when dealing with mutually exclusive events, the probability of 𝐴 intersection 𝐵 is equal to zero. The probability of 𝐴 union 𝐵 is therefore equal to one-sixth plus three-sixths minus zero. This is equal to four-sixths, which once again simplifies to two-thirds. This confirms that the probability that the selected singer was a tenor singer or soprano singer is two-thirds.
