Class 11 math (India)
Select problems from exercise 16.3
Solutions to some problems of NCERT exercise.
In this article we will look at solutions of a few selected problems from exercise 16.3 of NCERT.
A fair coin is tossed four times, and a person wins Re for each head and loses Rs for each tail that turns up.
From the sample space calculate how many different amounts of money you can have after four tosses and the probability of having each of these amounts.
On each toss the person may either win or lose. The final amount the person gets depends on the number of times he wins or loses.
There are a total of possible cases. Below and denote a win and loss respectively.
Case : wins.
Outcome is . Here the person wins Rs in total.
Case : wins and loss.
Outcomes are , , and . Here the person wins Rs in total.
Case : wins and losses.
Outcomes are , etc. Here the person loses Re in total.
Try it out
The bigger question is, how many outcomes are possible for this case?
We can list out all the outcomes but a shortcut would be to use permutation theory.
Basically, here we are arranging objects, identical and identical in a line. Number of ways to do that is .
Case : win and losses.
Try it out
Using permutation theory, find out the number of outcomes in this case.
There are outcomes: , , and . Here the person loses Rs in total.
Case : losses.
Outcome is . Here the person loses Rs in total.
Now how do we find the probability for each case?
Total number of outcomes is the sum of number of outcomes in each case .
Required probabilities are
|Case||Result||No. of favourable outcomes||Probability|
In a lottery, a person choses six different natural numbers at random from to , and if these six numbers match with the six numbers already fixed by the lottery committee, he wins the prize. What is the probability of winning the prize in the game.
[Hint: order of numbers is not important.]
What is the number of possible outcomes in this case?
The person can pick any combination of numbers from numbers. Number of possible choices is .
Out of these, only is the winning combination. The person wins if he picks this exact combination.
Probability of winning .
Events and are such that . State whether and are mutually exclusive.
Mutually exclusive events can never happen simultaneously. In other words, if events and are mutually exclusive, .
How can we convert this into an equation about probability of ?
Given . So
Because , and are not mutually exclusive events.
The probability that a student will pass the final examination in both English and Hindi is and the probability of passing neither is . If the probability of passing the English examination is , what is the probability of passing the Hindi examination?
Let us make a Venn diagram of probabilities using given data.
Because sum of all probabilities should equal , we have