Solved Examples

How to Use This Section

This is your practice bank for Probability. Every answer is (favourable outcomes)/(total outcomes) with equally likely outcomes — so the whole skill is careful counting, plus the complement $P(\overline{E})=1-P(E)$ when "not / at least / none" appears. Keep the standard totals ready: coin 2, two coins 4, die 6, two dice 36, deck 52.

Example 1: A die is thrown. $P(\text{odd})$? Odd $\{1,3,5\}$: $\dfrac36=\dfrac12$.

Example 2: $P(\text{a tail})$ when one coin is tossed? $\dfrac12$.

Example 3: Two coins tossed. $P(\text{no head})$? Only $TT$: $\dfrac14$.

Example 4: Two coins tossed. $P(\text{at least one tail})$? $\{HT,TH,TT\}$: $\dfrac34$.

Example 5: A die is thrown. $P(\text{a number} \le 2)$? $\{1,2\}$: $\dfrac26=\dfrac13$.

Example 6: Two dice thrown. $P(\text{sum}=9)$? $\{(3,6),(4,5),(5,4),(6,3)\}$ = 4 ways: $\dfrac4{36}=\dfrac19$.

Example 7: Two dice thrown. $P(\text{sum is a prime})$? Prime sums 2,3,5,7,11 with ways $1+2+4+6+2=15$: $\dfrac{15}{36}=\dfrac5{12}$.

Example 8: Two dice thrown. $P(\text{both odd})$? Odd faces $\{1,3,5\}$: $3\times3=9$ ways: $\dfrac9{36}=\dfrac14$.

Example 9: One card drawn from 52. $P(\text{a heart})$? $\dfrac{13}{52}=\dfrac14$.

Example 10: $P(\text{a black card})$? $\dfrac{26}{52}=\dfrac12$.

Example 11: $P(\text{a face card or an ace})$? $12+4=16$: $\dfrac{16}{52}=\dfrac4{13}$.

Example 12: $P(\text{a card that is neither a heart nor a king})$? Hearts 13, kings 4, but king of hearts counted twice, so $13+4-1=16$ excluded; favourable $52-16=36$: $\dfrac{36}{52}=\dfrac9{13}$.

Example 13: All the red cards are removed; a card is drawn from the remaining 26. $P(\text{a king})$? Only 2 black kings remain: $\dfrac{2}{26}=\dfrac1{13}$.

Example 14: A bag has 6 red, 4 white, 5 blue balls. $P(\text{red})$? Total 15: $\dfrac{6}{15}=\dfrac25$.

Example 15: For the same bag, $P(\text{not blue})$? $1-\dfrac{5}{15}=\dfrac23$.

Example 16: A number chosen from 1 to 50. $P(\text{a multiple of 5})$? $\{5,10,\dots,50\}=10$: $\dfrac{10}{50}=\dfrac15$.

Example 17: A number from 1 to 30. $P(\text{a multiple of 3 or 5})$? Multiples of 3: 10; of 5: 6; of 15: 2; so $10+6-2=14$: $\dfrac{14}{30}=\dfrac{7}{15}$.

Example 18: A spinner has 12 equal sectors numbered 1–12. $P(\text{a number} > 8)$? $\{9,10,11,12\}$: $\dfrac{4}{12}=\dfrac13$.

Example 19: If $P(E)=0.05$, what is $P(\overline{E})$? $1-0.05=0.95$.

Example 20: A box has cards numbered 1 to 100. $P(\text{a perfect square})$? $1,4,9,\dots,100 = 10$: $\dfrac{10}{100}=\dfrac1{10}$.

Example 21: Two dice thrown. $P(\text{product is 12})$? $\{(2,6),(6,2),(3,4),(4,3)\}=4$: $\dfrac4{36}=\dfrac19$.

Example 22: A card drawn from 52. $P(\text{a red face card})$? Red face cards $=6$: $\dfrac{6}{52}=\dfrac3{26}$.

Example 23: A die thrown twice. $P(\text{a doublet of even numbers})$? $\{(2,2),(4,4),(6,6)\}=3$: $\dfrac3{36}=\dfrac1{12}$.

Example 24: A bag has 3 red and 2 black balls. Two more red balls are added. Now $P(\text{red})$? Reds $5$, total $7$: $\dfrac57$.

Example 25: From 1 to 100, $P(\text{a number divisible by 4})$? $4,8,\dots,100 = 25$: $\dfrac{25}{100}=\dfrac14$.

Example 26: One coin tossed three times. $P(\text{exactly two heads})$? Sample space 8; favourable $\{HHT,HTH,THH\}=3$: $\dfrac38$.

Example 27: Two dice thrown. $P(\text{sum} \ge 10)$? Sums 10,11,12 with $3+2+1=6$ ways: $\dfrac6{36}=\dfrac16$.

Example 28: A letter is chosen from the word "PROBABILITY". $P(\text{the letter B})$? The word has 11 letters, B appears twice: $\dfrac{2}{11}$.

Example 29: A bag has balls numbered 1 to 9. $P(\text{an odd number})$? Odd $\{1,3,5,7,9\}=5$: $\dfrac59$.

Example 30: Mixed — full reasoning

A card is drawn from a well-shuffled deck of 52. Find the probability that it is (i) a queen, (ii) not a queen, (iii) a black queen, (iv) a queen or a king.

Solution:

1. (i) 4 queens: $\dfrac{4}{52}=\dfrac1{13}$.
2. (ii) $1-\dfrac1{13}=\dfrac{12}{13}$.
3. (iii) 2 black queens: $\dfrac{2}{52}=\dfrac1{26}$.
4. (iv) $4+4=8$: $\dfrac{8}{52}=\dfrac2{13}$.

Final Answer: $\dfrac1{13},\ \dfrac{12}{13},\ \dfrac1{26},\ \dfrac2{13}$.

Takeaway: Keep the denominator 52; the complement handles the "not" case instantly.