Question

Troy has two $1, two $2, and ten $3 gift certificates. If he selects three bills in succession, what is the probability that a $1 gift certificate is selected, then a $ 3 gift certificate and then a $2 gift certificate if replacement does not take place? and is the answer dependant or independant?

Answer 1

Answer:

$\dfrac{5}{273}$

They are dependent events.

Step-by-step explanation:

Given:

Number of $1 gift certificates = 2

Number of $2 gift certificates = 2

Number of $3 gift certificates = 10

To find:

Probability that a $1 gift certificate is selected, then a $3 and then a $2 certificate is selected = ?

Solution:

First of all, let us learn about the formula of probability of any event E:

$P(E) = \dfrac{\text{Number of favorable cases}}{\text {Total number of cases}}$

Finding probability of choosing $1 gift certificate in the first turn:

Number of favorable cases or number of $1 gift certificates = 2

Total number of cases or total number gift certificates = 2 + 2 + 10 = 14

So, required probability = $\frac{2}{14} \Rightarrow \frac{1}{7}$

Finding probability of choosing $3 gift certificate in the 2nd turn:

Number of favorable cases or number of $1 gift certificates = 10

Now, one gift certificate is already chosen,

So, Total number of cases or total number gift certificates = 14 - 1 = 13

So, required probability = $\frac{10}{13}$

Finding probability of choosing $2 gift certificate in the 3rd turn:

Number of favorable cases or number of $2 gift certificates = 2

Now, two gift certificates is already chosen,

So, Total number of cases or total number gift certificates = 14 - 2 = 12

So, required probability = $\frac{2}{12} \Rightarrow \frac{1}{6}$

Probability that a $1 gift certificate is selected, then a $3 and then a $2 certificate is selected:

$\dfrac{1}{7}\times \dfrac{10}{13}\times \dfrac{1}{6}\\\Rightarrow \dfrac{5}{273}$

They are dependent events because total number of cases and total number of favorable cases depend on the previous chosen gift certificate.

So, the answer is:

$\dfrac{5}{273}$

They are dependent events.