Probability and Chance

Probability is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, which is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems. In this article we shall discuss about probability and chance problems.(Source: wikipedia)

Probability and chance example problem

Example 1:

If the odds in favor of an event be 3/5 find the chance of the occurrence of the event.

Solution:

Let the given event be E and let P(E)=x then

Odds in favor of E =`(P(E))/(1-P(E))`

= `(P(E))/(1-P(E))` =`3/5`

= `(x)/(1-x)` =`3/5`

5x=3-3x

8x=3

x= `3/8`

Therefore required probability = `3/8`

Example 2:

There dice are thrown together. Find the chance of getting a total of at least 6.

Solution:

In throwing 3 dice together the number of all possible outcomes is (6x6x6)=216

Let E = event of getting a total of at least 6.

Then, E = event of getting a total of less than 6.

= event of getting a total of 3 or or 5.

={(1, 1, 1),(1,1,2),(1,2,1),(2,1,1)(1,1,3),(1,3,1),(3,1,1),(1,2,2),(2,1,2)(2,2,1)}

Now , n(`barE` )=10= P(not E)=P(`barE` )=`(n(barE))/(n(S))` =`10/216` =`5/108`

=P(E)=1-P(not E)=1-`5/108` =`103/108`

Hence the required probability is `103/108`

Example 3:

A bag contains 12 red and 15 white balloon. One ball is drawn at random. Find the chance that the ball drawn is red

Solution

Total number of balls=(12+15)=27

Let S be the sample space. Then

n(S)= number of ways of selecting 1 balloon out of 27=27

Let E be the event of drawing a red balloon. Then

n(E)= number of ways of selecting 1 red balloon out of 12=12

Therefore P(getting a /red balloon)=P(E)=`(n(E))/(n(S))` =`12/27` = `4/9`

Probability and chance practice problem

Problem 1:

The odds in favor of the outcomes of an event are 8:13 find the chance that the event will occur

Answer:

`8/21`

Problem 2:

If the odds against the occurrences of an event be 4:7 find the chance of the occurrences of the event?

Answer:

`7/11`