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PROBABILITY

Posted by : Allan_Dell on Monday, July 14, 2014 | 7:25 PM

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: (Pn + Pm) = 1Pn5 + 5 Pn4Pm + 10Pn3Pm2 + 10Pn2Pm35 PnPm4  + 1Pm5

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10:49 PM MST

Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is expressed from zero to one. Probability has been introduced in Maths to predict how likely events are to happen. The meaning of probability is basically the extent to which something is likely to happen. This is the basic probability theory which is also used in the probability distribution, where you will learn the possibility of outcomes for a random experiment. To find the probability of a single event to occur, first, we should know the total number of possible outcomes. When I started reading about probability in 10th class, I had a lot of difficulty in the beginning. But after practice, I got to know about this chapter. Thanks for this article.

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