Question

1. there is a set of 12 cards numbered 1 to 12. a card is chosen at random. event a is choosing a number less than 6. event b is choosing a number divisible by 3. a diagram is given below.

determine the probability of the following. leave your answer in simplified fraction form. (1 point total)
a. ( p(a) ) b. ( p(b) )
c. ( p(acup b) ) d. ( p(acap b) )

1. there are 6 different colored pens in a box. each pen has a unique color. in how many orders can 4 pens be chosen? in other words, what is the number of permutations of picking 4 pens from the box? (lesson 19.2) (1 point)

a. 720
b. 360
c. 30

Explanation:

Step1: Calculate \(P(A)\)

The formula for probability is \(P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\).
For event \(A\) (choosing a number less than \(6\)), the favorable outcomes are \(1,2,3,4,5\). So the number of favorable outcomes \(n(A)=5\).
Total number of outcomes \(n(S) = 12\).
So \(P(A)=\frac{5}{12}\).

Step2: Calculate \(P(B)\)

For event \(B\) (choosing a number divisible by \(3\)), the favorable outcomes are \(3,6,9,12\). So the number of favorable outcomes \(n(B) = 4\).
Using the probability formula \(P=\frac{n(B)}{n(S)}\), we get \(P(B)=\frac{4}{12}=\frac{1}{3}\).

Step3: Calculate \(P(A\cup B)\)

The formula for \(P(A\cup B)\) is \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\).
We know \(P(A)=\frac{5}{12}\), \(P(B)=\frac{4}{12}\).
For \(A\cap B\) (number is less than \(6\) and divisible by \(3\)), the favorable outcome is \(3\), so \(P(A\cap B)=\frac{1}{12}\).
Substituting the values: \(P(A\cup B)=\frac{5 + 4-1}{12}=\frac{8}{12}=\frac{2}{3}\).

Step4: Calculate \(P(A\cap B)\)

As mentioned above, for \(A\cap B\) (number is less than \(6\) and divisible by \(3\)), the favorable outcome is \(3\).
Using the probability formula \(P=\frac{n(A\cap B)}{n(S)}\), with \(n(A\cap B) = 1\) and \(n(S)=12\), we get \(P(A\cap B)=\frac{1}{12}\).

Step5: Solve the permutation problem (Question 4)

The permutation formula is \(P(n,r)=\frac{n!}{(n - r)!}\), where \(n = 6\) (total number of pens) and \(r=4\) (number of pens chosen).
\(P(6,4)=\frac{6!}{(6 - 4)!}=\frac{6!}{2!}=\frac{6\times5\times4\times3\times2!}{2!}=6\times5\times4\times3=360\).

Answer:

a. \(\frac{5}{12}\)
b. \(\frac{1}{3}\)
c. \(\frac{2}{3}\)
d. \(\frac{1}{12}\)

1. B. 360