Question

Consider a pack of standard playing cards. A card is drawn at random from the pack. A card is said to be Royal if it is a Jack, Queen or King.. Let A be the event that the card is a king. Let B be the event that the card is a heart. Lct C be thc event that the card is royal. Lct D be the event that the card is black. Calculate the following probabilities. (2) P(AnB) (3) P(C) (4) P(AUC) (S) P(BnD) (6) P(AUD) 2. Stores sell glass cup with box, 20s in cach box. The probability that there is 0,1,or 2 defective item in cach box is 0.8, 0.1 or 0.1 respectively. A customer picks out a box and choose 4 to check from it. If all results are okay, then he buy this box. Find the probability that there is one defective item in the box. 3. Consider two fair dice A and B. Die A is six-sided and is numbered 1 through to 6 whilst dic B is four-sided and is numbered 1 through to 4. Both dice are rolled. Let X-AtB. Find the probability mass function and cumulative distribution function of X

Answer 1

Given a pack of cards. The following events are defined

$A=\left \{ \textup{The card is a King }\right \}\\ B=\left \{ \textup{The card is a Heart }\right \}\\ C=\left \{ \textup{The card is Royal }\right \}\\ D=\left \{ \textup{The card is black }\right \}\\$

1) The probability $P\left ( A \right )$ is nothing but the ratio of 4 possible Kings to the total number of playing cards.

${\color{Blue} P\left ( A \right )=\frac{4}{52}=1/13}$

2) The probability $P\left ( AB \right )$ is the ratio of the number of King of Heart to the total number of playing cards.

${\color{Blue} P\left ( A B\right )=\frac{1}{52}}$

3) The probability of a Royal ( the card is J,Q, K) is

${\color{Blue} P\left ( C\right )=\frac{4+4+4}{52}=3/13}$

4) The probability of a Royal ( the card is J,Q, K) or the card is a King is ( $\left |\textup{AC} \right |=4$ )

$P\left ( A\cup C\right )= P\left ( A\right )+ P\left ( C\right )- P\left ( A\cap C\right )\\ P\left ( A\cup C\right )= 4/52+ 12/52- 4/52\\ {\color{Blue} P\left ( A\cup C\right )= 3/13}$

5) The event $B\cap D$ represents the event the card is black and the cards is of Heart. Hence $B\cap D=\phi$

${\color{Blue} P\left ( B\cap D\right )=0}$

We are required to solve only 4 parts. Please post the remaining questions as another post.