$17$ cards numbered $1, 2, 3, \dots, 16, 17$ are put in a box and mixed thoroughly. One person draws a card from the box. Find the probability that the number on the card is (i) odd (ii) a prime (iii) divisible by 3 (iv) divisible by 3 and 2 both.

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Solution

Step 1: Find the probability that the number on the card is odd:
The seventeen cards are numbered from $1, 2, 3, . . . 16, 17$
The odd numbers in the cards are $1, 3, 5, 7, 9, 11, 13, 15, 17$
Thus, the number of favorable outcomes for the given event is $9$
The formula for probability will be,
$Probability= \frac{Numberoffavorableoutcomes}{Totalnumberofoutcomes}$
Thus, the probability will be,
$Probability= \frac{9}{17}$
Step 2: Find the probability that the number on the card is prime:
The seventeen cards are numbered from $1, 2, 3, . . . 16, 17$
The prime numbers in the cards are $2, 3, 5, 7, 11, 13, 17$
Thus, the number of favorable outcomes for the given event is $7$
The formula for probability will be,
$Probability= \frac{Numberoffavorableoutcomes}{Totalnumberofoutcomes}$
Thus, the probability will be,
$Probability= \frac{7}{17}$
Step 3: Find the probability that the number on the card is divisible by 3:
The seventeen cards are numbered from $1, 2, 3, . . . 16, 17$
The numbers in the cards which are divisible by three are $3, 6, 9, 12, 15$
Thus, the number of favorable outcomes for the given event is $5$
The formula for probability will be,
$Probability= \frac{Numberoffavorableoutcomes}{Totalnumberofoutcomes}$
Thus, the probability will be,
$Probability= \frac{5}{17}$
Step 4: Find the probability that the number on the card is divisible by 3 and 2 both:
The seventeen cards are numbered from $1, 2, 3, . . . 16, 17$
The numbers in the cards which are divisible by three and two both are $6, 12$
Thus, the number of favorable outcomes for the given event is $2$
The formula for probability will be,
$Probability= \frac{Numberoffavorableoutcomes}{Totalnumberofoutcomes}$
Thus, the probability will be,
$Probability= \frac{2}{17}$
Hence, the probability that the number on the card is
(i) odd $= \frac{9}{17}$
(ii) a prime $= \frac{7}{17}$
(iii) divisible by 3 $= \frac{5}{17}$
(iv) divisible by 3 and 2 both $= \frac{2}{17}$

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