there are 7 people fishing at lake connor: 5 have fishing licenses, and 2 do not. an inspector chooses two…

there are 7 people fishing at lake connor: 5 have fishing licenses, and 2 do not. an inspector chooses two of the people at random. what is the probability that both people have licenses? write your answer as a fraction in simplest form.

there are 7 people fishing at lake connor: 5 have fishing licenses, and 2 do not. an inspector chooses two of the people at random. what is the probability that both people have licenses? write your answer as a fraction in simplest form.

Answer

Explanation:

Step1: Calculate probability of first - person having a license

The total number of people is $n = 7$, and the number of people with licenses is $m = 5$. The probability that the first - person chosen has a license is $P_1=\frac{5}{7}$.

Step2: Calculate probability of second - person having a license given first has a license

After choosing one person with a license, there are $n_1=6$ people left and $m_1 = 4$ people with licenses left. So the probability that the second - person chosen has a license given that the first one had a license is $P_2=\frac{4}{6}$.

Step3: Calculate the probability that both have licenses

By the multiplication rule of probability for dependent events, the probability that both people have licenses is $P = P_1\times P_2=\frac{5}{7}\times\frac{4}{6}=\frac{20}{42}=\frac{10}{21}$.

Answer:

$\frac{10}{21}$