Of the 10,000 students at a certain university, 7000 have Visa cards, 6000 have MasterCards, and 5000 have both. Suppose that a student is randomly selected.

a. What is the probability that the selected student has a Visa card?

b. What is the probability that the selected student has both cards?

c. Suppose you learn that the selected individual has a Visa card (was one of the 7000 with such a card). Now what is the probability that this student has both cards?

d. Are the events has a Visa card and has a MasterCard independent? Explain.

e. Answer the question posed in Part (d) if only 4200 of the students have both cards.

#### Solution

Given that

$P(\text{Visa Card}) = \frac{7000}{10000}=0.7$.

$P(\text{Master Card}) = \frac{6000}{10000}=0.6$

$P(\text{Visa and Master Card}) = \frac{5000}{10000}=0.5$

(a) The probability that the selected student has a Visa card is

`

$$

\begin{aligned}

P(\text{Visa Card}) &= \frac{7000}{10000}\

&=0.7.

\end{aligned}

$$

(b) The probability that the selected student has both cards is

` $$ \begin{aligned} P(\text{Visa and Master Card}) &= \frac{5000}{10000}\\ &=0.5 \end{aligned} $$ `

(c) The probability that selected student has both the cards given that he has Visa card is

` $$ \begin{aligned} P(\text{Visa and Master Card}|\text{Visa Card}) &= P(\text{Visa card} \cap \text{Master Card}|\text{Visa Card})\\ &=\frac{P(\text{Visa card} \cap \text{Master Card})}{P(\text{Visa Card})}\\ &= \frac{\frac{5000}{10000}}{\frac{7000}{10000}}\\ &=\frac{5000}{7000}\\ &= 5/7=0.71429 \end{aligned} $$ `

(d)

` $$ \begin{aligned} P(\text{Visa Card})\times P(\text{Master Card}) &= 0.7\times 0.6=0.42\\ &\neq 0.5 = P(\text{Visa and Master Card}) \end{aligned} $$ `

Hence, the events has a Visa card and has a MasterCard are not independent.

(e)

Given that

` $$ \begin{aligned} P(\text{Visa and Master Card}) &=\frac{4200}{10000}\\ =0.42 \end{aligned} $$ `

` $$ \begin{aligned} P(\text{Visa Card})\times P(\text{Master Card}) &= 0.7\times 0.6=0.42\\ &= 0.42 = P(\text{Visa and Master Card}) \end{aligned} $$ `

Hence, the events has a Visa card and has a MasterCard are independent.

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators