The probability that a student chosen at random from a class has ginger hair is [tex]$\frac{5}{14}$[/tex]. The probability that they have blonde hair is [tex]$\frac{9}{28}$[/tex]. What is the probability that a student chosen at random from the class has either ginger hair or blonde hair? Give your answer as a fraction.
Answer (2)
Define the probabilities: P ( G ) = 14 5 and P ( B ) = 28 9 .
Use the formula for mutually exclusive events: P ( G ∪ B ) = P ( G ) + P ( B ) .
Substitute and find a common denominator: P ( G ∪ B ) = 14 5 + 28 9 = 28 10 + 28 9 .
Add the fractions: P ( G ∪ B ) = 28 19 .
Explanation
Understand the problem and provided data We are given the probability of a student having ginger hair as 14 5 and the probability of a student having blonde hair as 28 9 . We want to find the probability that a student has either ginger or blonde hair. We assume that a student cannot have both ginger and blonde hair at the same time, so the events are mutually exclusive.
Define probabilities and the formula Let P ( G ) be the probability of a student having ginger hair, and P ( B ) be the probability of a student having blonde hair. We are given P ( G ) = 14 5 and P ( B ) = 28 9 . We want to find P ( G ∪ B ) , the probability of a student having either ginger or blonde hair. Since the events are mutually exclusive, we have P ( G ∪ B ) = P ( G ) + P ( B ) .
Substitute the values Substitute the given values: P ( G ∪ B ) = 14 5 + 28 9 . To add these fractions, we need a common denominator. The least common multiple of 14 and 28 is 28. So we rewrite 14 5 as 28 10 .
Add the fractions Now we have P ( G ∪ B ) = 28 10 + 28 9 . Adding the fractions, we get P ( G ∪ B ) = 28 10 + 9 = 28 19 .
State the final answer Therefore, the probability that a student has either ginger or blonde hair is 28 19 .
Examples
This type of probability calculation is useful in genetics to determine the likelihood of offspring inheriting certain traits. For example, if you know the probability of inheriting a gene for ginger hair and a gene for blonde hair, you can calculate the probability of inheriting either of those hair colors. This assumes that the genes for ginger and blonde hair are mutually exclusive, meaning a person can't inherit both at the same time. The calculation helps predict the distribution of traits in a population.
The probability that a student chosen at random from the class has either ginger hair or blonde hair is 28 19 . This was calculated by using the probabilities given and adding them together after converting to a common denominator. The events are assumed to be mutually exclusive, meaning a student cannot have both hair colors at the same time.
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