Question

Solve the following by using the binomial formula.

1.When rolling a die 100 times, what is the probability rolling a 4 exactly 25 times?

2.       A teacher developed a 5- item multiple choice question with four options in each item. What is the probability of that a certain student who randomly selects his answers will get exactly 4 correct answers?

Answer 1

$1:\\p=\frac{1}{6}\\X\sim Bin\left( 100,\frac{1}{6} \right) \\P\left( X=25 \right) =C_{100}^{25}\cdot \left( \frac{1}{6} \right) ^{25}\left( 1-\frac{1}{6} \right) ^{100-25}=0.00982588\\2:\\p=0.25\\X\sim Bin\left( 5,0.25 \right) \\P\left( X=4 \right) =C_{5}^{4}\cdot 0.25^4\cdot \left( 1-0.25 \right) ^{5-4}=0.0146484$