Probability

Example 1:If a card is drawn at random from an ordinary deck of 52 cards, find the probability that it is a spade or an even numbered card.

Example 2:The probability that a patient entering UMC Hospital will consult a physician is 0.7, that he/she will consult a dentist is 0.5 and that he/she will consult a physician or a dentist or both is 0.9. What is the probability that a patient entering the hospital will consult both a physician and a dentist?

Example 3:Three dice are thrown, what is the probability that the 3 dice show even numbers?

Example 4:The probability that a certain movie will get an award for good acting is 0.16, the probability that it will get an award for good directing is 0.27, and the probability that it will get awards for both is 0.11. What is the probability that the movie will get only one of the two awards?

Example 5:In a high school graduating class of 100 students, 54 studied mathematics, 69 studied history, and 35 studied both mathematics and history. If one of this students is selected at random, find the probability that: (a) the student took mathematics or history but not both. (b) a student did not take either of these subjects. (c) the student took history but not mathematics.

Example 6:There are two flocks of birds, one below the other. The lower flock says “If one of us goes up there, you will double our number. But if one of you goes down here, we will be equal in number.” If a bird is chosen from these flocks, what is the probability that the bird chosen at random is from the lower flock?

Example 7:In a poker hand consisting of 5 cards, find the probability of holding 4 hearts and 1 club.

Conditional ProbabilityThe probability of an event B occurring

when it is known that some event A has occurred is called a conditional probability. It is denoted by the symbol P(BІA) which is usually read as “the probability that B occurs given that A occurs” or simply “the probability of B given A.”

Conditional Probability

Definition:Let A and B be events such that P(A) ≠ 0. The conditional probability of B, given A, denoted by P(BІA), is given by

APBAP

ABP

Example 1:A card is drawn from a standard deck. Suppose we are told that the card picked is a spade. What is the probability that the card drawn is the ace of spades?

Example 2:A single fair die is rolled once. What is the probability that the number obtained is less than 4 knowing that odd number is rolled.

Example 3:The probability that a regular scheduled flight departs on time is P(D) = 0.83; the probability that it arrives on time is P(A) = 0.82; and the probability that it departs and arrives on time is P(DA) = 0.78. Find the probability that a plane (a) arrives on time given that it departed on time, and (b) departed on time given that it has arrived on time.

Example 4:Consider the population of adults in small town who have completed the requirements for a college degree. We shall categorize them according to gender and employment status:

One of these individuals is to be selected at random. Consider the following events:

M: a man is chosenE: the chosen is employed

Employed Unemployed Total

Male 460 40 500

Female 140 260 400

Total 600 300 900

Example 5:Consider the population of adults in small town who have completed the requirements for a college degree. We shall categorize them according to gender and employment status:

One of these individuals is to be selected at random. Consider the following events:

F: a female is chosenU: the chosen is unemployed

Employed Unemployed Total

Male 460 40 500

Female 140 260 400

Total 600 300 900

Independent Events

Two events are considered to be independent if the occurrence or non-occurrence of one has no influence of the other.

Independent Events

Definition:Two events are said to be independent if any one of the following conditions are satisfied:

Otherwise, A and B are dependent.

0 if

0 if

APBPBAP

BPAPABP

Multiplicative RulesIf events A and B are dependent,

then:

If events A and B are independent, then

ABPAPBAP

BPAPBAP

Example 1:The probability that Jack will correctly answer the toughest question in an exam is 1/4. The probability that Rose will correctly answer the same question is 4/5. Find the probability that both will answer the question correctly, assuming that they do not copy from each other.

Example 2:On example 1, find the probability that (a) Jack will get an incorrect answer and Rose will get the correct answer. (b) Jack will get the correct answer and Rose will get an incorrect answer. (c) both will get an incorrect answer.

Example 3:Suppose that we have a fuse box containing 20 fuses, of which 5 are defective. If two fuses are selected at random and removed from the box in succession without replacing the first, what is the probability that both fuses are defective?

Example 4:On example 3, what is the probability that (a) the first one is not defective and the second is defective. (b) the first one is defective and the second one is defective. (c) both are not defective.

Exercise 1:Three cards are drawn in succession, without replacement, from an ordinary deck of playing cards. Find the probability that the first card is a red ace, the second card is a 10 or a jack and the third card is greater than 3 but less than 7.

Exercise 2:A purse contains 3 1-peso coins and 2 5-peso coins. Another coin purse contains 2 1-peso coins and 5 5-peso coins . What is the probability that if a coin is selected at random, the coin is 5 peso? (page 135, no. 1)

Exercise 3:Box 1 contains 6 good and 2 defective light bulbs. Box 2 contains 3 good and 3 defective light bulbs. A box is selected at random and a light bulb is chosen from it. Find (a) the probability that the light bulb is defective. (b) the probability that the light bulb came from Box 2 given that it is defective.

Exercise 4:A box contains 6 red and 4 black balls. Two balls from the box are drawn one at a time without replacement. What is the probability that the second ball is red if it is known that the first is red? (page 139, no. 11)

Exercise 5:A hospital spokesperson reported that 4 births had taken place at the DLS-UMS during the last 24 hours. Find the following probabilities: (a) P(A) = that 2 boys and 2 girls are born. (b) P(B) = no boys are born. (c) P(C) at least one boy is born. (d) P(AІC) (e) P(BІC) (f) Are A and C mutually exclusive? Are A and C independent? (g) Are B and C mutually exclusive? Are B and C independent? (page 145, no. 25)