PROBABILITY

PROBABILITY

     -  a game of chance.

     - is a possibility or chance to be true to happen. (Calmorin et.al)

• If an event can succeed in s ways and fail in f ways, then the probability of failure are as follows:

(Probability of success and failure):

P(s) = s / (s+f) ; for success

P(f) = f / (s+f) ; for failure

Problem:

A box contains 3 base balls, 7 softballs, and 11 tennis balls. what is the probability that a ball selected at random will be tennis ball?

Answer: P(tennis balls) = s / (s+f) = 11 / [11 + (3+7)] = 11/21

Problem:

Two cards are drawn at random from a standard deck of 52 cards. what is the probability that both cards are hearts?

Answer: 1/17 (why?)

Problem:

A collection of 15 transistors contains 3 that are defective. If two transistors are selected by random, what is the probability that at least 1 of them is good? What is the probability of selecting at least one good transistors?

Answer: 1/35 (why?)
Answer: 34/35 (why?)


INDEPENDENT EVENTS

If two events, A and B, are independent, then the probability of both events occurring is:

• P(A and B) = P(A) x P(B)

Problem:

Find the probability of getting a sum of 7 on the first throw of two dice and s a sum of 4 on the second throw?

Answer: 1/72

Problem:

A  new phone is being installed at Smith residence. Find the probability that the final three digits in the telephone number will be even.

Answer:

P(any digit being even) = 5/10 = 1/2
P(final three being even) = 1/2 * 1/2 * 1/2 = 1/8


DEPENDENT EVENTS

If two events, A and B, are dependent, then the probability of both events occurring is:

• P(A and B) = P(A) x P(B following A) 

Problem:

There are 5 red, 3 blue, and 7 black marbles in a bag. Three marbles are chosen without replacement. Find the probability of selecting a red one, then a blue one, and then a red one.

Answer: P(red,blue,red) = 5/15 * 3/14 * 4/13 = 2/91

MUTUALLY EXCLUSIVE EVENTS

If two events, A and B, are mutually exclusive, then the probability of both events occurring is:

• P(A and B) = P(A) + P(B) 

Problem:

Find the probability of a sum of 6 or a sum of 9 on a single throw of two dice.

Answer:
P(sum of 6) = 5/36
P(sum of 9) = 4/36

Then P(A and B) = P(A) + P(B) = 5/36 + 4/36 = 1/4


INCLUSIVE EVENTS

If two events, A and B, are exclusive, then the probability of both events occurring is:

• P(A and B) = P(A) + P(B) - P(A and B)

Problem:

A letter is picked up at random from the English Alphabet. Find he probability that the letter is contained in the word house or in the word phone.

Answer: Let A be a letter from the word house, and B for phone.

P(A) = 5/26

P(B) = 5/26

P(A and B) = 3/26

Then, P(A or B) = 5/26 + 5/26  - 3/26 = 7/26

Problem:

A committee of five people is to be selected from a group of 6 men and 7 women. what is the probability that the committee will have at least 3 men?

Answer: 59/143


CONDITIONAL PROBABILITY

The conditional probability of event A, given event B, is found to be;

• P(A/B) = P(A and B) / P(B); P(B) not = to zero

A pair of dice are thrown. Find the probability that the numbers of the dice match given that thier sum is greater than 7.

P(B) = 15 / 36

P(A) = 3 / 36

Answer: P(A/B) = (3 / 36) / (15 / 36) = 1 / 5

BINOMIAL THEOREM AND PROBABILITY

  A binomial experiment exists if and only if the following conditional occur.

•         The experiment consists of n identical trials.
•         Each trial results in one of two outcomes.
•        The trials are independent.

Problem:

Suppose that 5 coins are tossed at the same time. What is the  probability that exactly 2 coins will show heads?

Answer: (P_n + P_m) = 1P_n⁵ + 5 P_n⁴P_m + 10P_n³P_m² + 10P_n²P_m³ + 5 P_nP_m⁴  + 1P_m⁵