Question

2a. if i randomly choose 1 student at random from the distribution, what is the probability he/she got an f? round to the tenth. 0.2 2b. if i randomly choose 1 student at random from the distribution, what is the probability he/she did not get an a or b? round to the tenth. 0.6 2c. if i randomly choose 1 student at random from the distribution, what is the probability he/she got an a and a b? 0 2d. if i randomly choose 3 students at random from the distribution, what is the probability all 3 got bs? round to the hundreth.

Explanation:

Step1: Calculate the probability of getting an F (for 2a)

The sum of all probabilities in a distribution is \(1\). Let \(P(F)\) be the probability of getting an \(F\).
We know \(P(A) = 0.1\), \(P(B)=0.3\), \(P(C) = 0.35\), \(P(D)=0.05\)
Using the formula \(P(A)+P(B)+P(C)+P(D)+P(F)=1\)
Substitute the values: \(0.1 + 0.3+0.35 + 0.05+P(F)=1\)
\(0.8+P(F)=1\)
\(P(F)=1 - 0.8=0.2\)

Step2: Calculate the probability of not getting an A or B (for 2b)

The probability of getting an \(A\) or \(B\) is \(P(A\cup B)=P(A)+P(B)\) (since \(A\) and \(B\) are mutually - exclusive events). \(P(A)+P(B)=0.1 + 0.3=0.4\)
Using the formula \(P(\text{not}(A\cup B))=1 - P(A\cup B)\)
\(P(\text{not}(A\cup B))=1-0.4 = 0.6\)

Step3: Calculate the probability of getting an A and B (for 2c)

A student cannot get both an \(A\) and a \(B\) simultaneously. For mutually - exclusive events \(A\) and \(B\) (a student can only have one grade), \(P(A\cap B) = 0\)

Step4: Calculate the probability of all 3 getting B's (for 2d)

Assume the selections are independent. If the probability of one student getting a \(B\) is \(P(B) = 0.3\)
For three independent events \(E_1\), \(E_2\), \(E_3\) (where \(E_i\) is the event that the \(i\) - th student gets a \(B\)), \(P(E_1\cap E_2\cap E_3)=P(E_1)\times P(E_2)\times P(E_3)\) (by the multiplication rule for independent events)
\(P = 0.3\times0.3\times0.3=0.027\approx0.03\)

Answer:

2a. \(0.2\)
2b. \(0.6\)
2c. \(0\)
2d. \(0.03\)