Question

Ten cards numbered through 1 to 10 are placed in a box, mixed up thoroughly and then 1 card is drawn randomly. If it is known that the number on the drawn card is more than 3, what is the probability that it is an even number?

Answer 1

Here we use the method of probability to find the probability of the given situation. We first find the number of cards that have numbers greater than the given number. Similarly we find the number of cards that have even numbers on them. Find the favorable and total number of outcomes and find the probability.

• A number is even if it is divided completely by 2, i.e. it can be written in the form 2n .
• Probability of an event is given by dividing number of favorable outcomes by total number
  of outcomes.

Complete step-by-step answer:
We are given cards numbered from 1 to 10 are put in a box
So, total number of cards in the box $= 10$
Now one card is drawn at random.
We find the number of cards that have a number that is greater than 3.
Since, we know there are 10 numbers from 1 to 10.
We count the numbers from 1 to 10 after the number 3(as 3 is not greater than 3).
Numbers greater than 3 are 4, 5, 6, 7, 8, 9 and 10
So, number of cards greater than 3 $= 7$
$\Rightarrow$Number of total outcomes is 7 ………….… (1)
Now we find the number of cards that are even from the obtained possibilities of card from
equation (1)
Since we now have cards 4, 5, 6, 7, 8, 9 and 10
From these observations the even numbers are 4, 6, 8 and 10
(As $4 = 2(2),6 = 2(3),8 = 2(4),10 = 2(5)$)
So, number of cards that have even numbers on them $= 4$
$\Rightarrow$Number of favorable outcomes is 4 ………..… (2)
Probability is given by number of favorable outcomes divided by total number of
observations.
Here the total number of observations is 7 and number of favorable outcomes is 4.
$\Rightarrow$ Probability $= \dfrac{4}{7}$

$\therefore$Probability of the card having an even number greater than 3 is $\dfrac{4}{7}$.

Note: Students might make mistake while calculating the numbers that are greater than 3 if
they include the number 3 along with the other numbers. Keep in mind it is not given greater
or equal to, else we would have chosen 3 along in the total observations.
Alternate method:
Number of total cards is 10.
We choose one card randomly.
Numbers greater than 3 are 4, 5, 6, 7, 8, 9 and 10
So, number of cards greater than 3 $= 7$
Probability of choosing a number greater than 3 from total 10 cards $= \dfrac{7}{{10}}$
………... (1)
Similarly, cards having even number greater than 3 are 4, 6, 8 and 10
So, number of cards having even number greater than 3 $= 4$
Probability of choosing an even number greater than 3 from total 10 cards $= \dfrac{4}{{10}}$ ……...… (2)
Since, we have to find the probability of card having even number greater than 3 from the
set of cards having numbers greater than 3.
Probability will be given by dividing equation (2) by equation (1)
$\Rightarrow$ Probability $= \dfrac{{\dfrac{4}{{10}}}}{{\dfrac{7}{{10}}}}$
Cancel same term in denominator of fractions in numerator and denominator.
$\therefore$ Probability $= \dfrac{4}{7}$