Question

Determine the probability p. for the event.
(I) An odd number appears in a single toss of a fair die.
(II) At least one head appears in two tosses of a fair coin.
(III) A king, 9 of hearts, or 3 of spades appears in drawing in a single card from a well shuffled ordinary deck of 52 cards.
(IV) The sum of 6 appears in a single toss of a pair of fair dice.

Answer 1

(I) Total possible outcomes of a single throw of a die are $S=\{1,2,3,4,5,6\}$
Out of which favorable i.e., odd outcomes are $=\{1,3,5\}$
Use probability
$=\frac{\text{Number of favorable outcomes}}{\text{Number of Total outcomes}}$

$\therefore \text{Probability}=\frac{3}{6}=\frac{1}{2}$

(II) When a fair coin is tossed two times, the sample space is
$S=\left\{\text{HH, HT, TH, TT}\right\}$

If at least one head appears then the favorable cases are HH, HT, TH

Probability $=\frac{\text{Number of favorable outcomes}}{\text{Number of total outcomes}}$

$\therefore \text{Probability}=\frac{3}{4}$

(III) Total possible outcomes are 52 cards
Favorable outcomes are 4 kings + 9 of hearts + 3 of spades $=4+1+1=6$

$\text{Probability}=\frac{\text{Number of favorable outcomes}}{\text{Number of total outcomes}}$

$\therefore \text{Probability}=\frac{6}{52}=\frac{3}{26}$

(IV) When a pair of dice is rolled, total number of outcomes $=6\times 6=36$

If sum is 6 then possible favorable events $=(1,5),(5,1),(2,4),(4,2)\text{and}(3,3)$

$\text{Probability}=\frac{\text{Number of favorable outcomes}}{\text{Number of total outcomes}}$

$\therefore \text{Probability}=\frac{5}{36}$