Q18(iii):
A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears (iii) a number divisible by 5.

Solution :

Given: A box contains 90 discs, numbered from 1 to 90. One disc is drawn at random.

To Find: The probability that the drawn disc bears a number divisible by 5.

Step 1: Determine the Total Number of Possible Outcomes

Since the discs are numbered from 1 to 90, the total number of discs in the box is 90. Therefore, the total number of possible outcomes ($n(S)$) is:

$n(S) = 90$

Step 2: Identify the Favorable Outcomes

Let $E$ be the event of drawing a disc with a number divisible by 5. The numbers between 1 and 90 that are divisible by 5 form an arithmetic progression (AP):

$5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90$

To find the count of these numbers, we use the formula for the $n^{th}$ term of an arithmetic progression: $a_n = a + (n - 1)d$.

Here, the first term ($a$) = 5, the common difference ($d$) = 5, and the last term ($a_n$) = 90.

$90 = 5 + (n - 1)5$

[Subtract 5 from both sides]

$85 = (n - 1)5$

[Divide both sides by 5]

$17 = n - 1$

[Add 1 to both sides]

$n = 18$

Thus, the number of favorable outcomes ($n(E)$) is 18.

Step 3: Calculate the Probability

The probability of an event $E$ is defined by the formula:

$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{n(E)}{n(S)}$

[Substituting the values obtained in Step 1 and Step 2]

$P(E) = \frac{18}{90}$

Step 4: Simplify the Fraction

[Divide both numerator and denominator by their greatest common divisor, which is 18]

$P(E) = \frac{18 \div 18}{90 \div 18}$

$P(E) = \frac{1}{5}$

Final Answer: The probability that the drawn disc bears a number divisible by 5 is $\frac{1}{5}$ or 0.2.

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Q18(iii): A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears (iii) a number divisible by 5.

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Q18(iii):
A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears (iii) a number divisible by 5.