Respuesta :
Sample space (N(S)) = 50
Let A be the event of getting a number divisible by 3
A = {3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48}
N(A) = 16
Therefore, the probability of a number divisible by 3 is
[tex]P(A)=\frac{N(A)}{N(S)}=\frac{16}{50}[/tex]Let B be the event of getting a number divisible by 4
B = { 4,8,12,16,20,24,28,32,36,40,44,48}
N(B) = 12
Therefore, the probability of a number divisible by 4 is
[tex]P\left(B\right)=\frac{N(B)}{N(S)}=\frac{12}{50}[/tex]Also, the intersection between A and B is
A n B = { 12,24,36,48}
Therefore,
[tex]P\left(A\cap B\right)=\frac{N(A\cap B)}{N(S)}=\frac{4}{50}[/tex]Hence, the probability that a number selected at random from the first 50 positive integers is exactly divisible by 3 or 4 will be
[tex]P\left(AUB\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)[/tex]Therefore,
[tex]\begin{gathered} P(A\cup B)=\frac{16}{50}+\frac{12}{50}-\frac{4}{50}=\frac{24}{50}=\frac{12}{25} \\ \therefore P(A\cup B)=\frac{12}{25} \end{gathered}[/tex]Hence, the answer is
[tex]\frac{12}{25}[/tex]