Since we are asked to find the probability of an event ocurring exactly 50 times, we can use the binomial formula:
[tex]P(X)=_nC_x\cdot p^x\cdot(1-p)^{n-x}[/tex]where n is the number of trials, x is the number of successes and p is the probability of success on an individual trial.
[tex]_nC_x=\frac{n!}{x!(n-x)!}[/tex]With this fomula, let's use the data we were given:
[tex]P(X)=_{300}C_{50}\cdot(0.17)^{50}\cdot(1-0.17)^{300-50}[/tex][tex]P(X)=\frac{300!}{50!(250!)}\cdot(0.17)^{50}\cdot0.83^{250}[/tex]Using a calculator or a computer to do this operation, we get:
[tex]P(X)\approx0.060969262[/tex]And so, round to the neares ten thousandth:
[tex]P(X)=0.0610[/tex]or about 6.10%.
