9th Class Math Chapter 13 Notes

Q1 — Dice labelled L,M,N,O,P,U: probability of a consonant

Find $P(\text{consonant})$.

1. Vowels among the labels: $O,U$ (2); consonants: $L,M,N,P$ (4).

Answer: $P=\frac{4}{6}=\frac{2}{3}$

Q2 — Dice: $P(3\text{ or }4)$ and $P(\text{not }3\text{ or }4)$

Find both probabilities.

1. $P(3\text{ or }4)=\frac{2}{6}=\frac{1}{3}$.
2. $P(\text{not})=1-\frac{1}{3}$.

Answer: $P(3\text{ or }4)=\frac{1}{3}$; $P(\text{not})=\frac{2}{3}$

Q3 — $P(\text{sum}\ge 4)$

Two fair dice. Find the probability the sum is at least $4$.

1. Complement: sum $<4$ means sum $=2$ (1 way) or $3$ (2 ways) — $3$ outcomes out of $36$.
2. $P(\ge 4)=1-\frac{3}{36}$.

Answer: $\frac{33}{36}=\frac{11}{12}$

Q4 — $P(\text{product between }5\text{ and }10)$

Find the probability.

1. Products $5$–$10$: $5\to (1,5),(5,1)$; $6\to (1,6),(6,1),(2,3),(3,2)$; $8\to (2,4),(4,2)$; $9\to (3,3)$; $10\to (2,5),(5,2)$.
2. Total favourable outcomes $=2+4+2+1+2=11$.

Answer: $\frac{11}{36}$

Q5 — $P(|difference|=4)$

Find the probability the difference between the two dice is $4$.

1. Pairs: $(1,5),(5,1),(2,6),(6,2)$ — $4$ outcomes.

Answer: $\frac{4}{36}=\frac{1}{9}$

Q6 — $P(\text{first}\ge 5\text{ and second}\ge 4)$

Find the probability.

1. First die $\in \{5,6\}$: $2$ options; second die $\in \{4,5,6\}$: $3$ options.
2. Favourable $=2\times 3=6$.

Answer: $\frac{6}{36}=\frac{1}{6}$

Q7 — $P(\text{vowel})$

MATHEMATICS has $11$ letters. Find $P(\text{vowel})$.

1. Vowels: $A,E,A,I$ — $4$ vowels out of $11$.

Answer: $\frac{4}{11}$

Q8 — $P(\text{consonant})$

Find $P(\text{consonant})$.

1. Consonants $=11-4=7$.

Answer: $\frac{7}{11}$

Q9 — $P(E)$

Find the probability of selecting E.

1. $E$ appears once.

Answer: $\frac{1}{11}$

Q10 — $P(A)$

Find the probability of selecting A.

1. $A$ appears twice.

Answer: $\frac{2}{11}$

Q11 — $P(\text{not }M)$

Find the probability.

1. $M$ appears twice: $P(M)=\frac{2}{11}$; $P(\text{not }M)=1-\frac{2}{11}$.

Answer: $\frac{9}{11}$

Q12 — $P(\text{not }T)$

Find the probability.

1. $T$ appears twice: $P(\text{not }T)=1-\frac{2}{11}$.

Answer: $\frac{9}{11}$

Q13 — $P(\text{the number }25)$

Find the probability.

1. Only one card shows $25$.

Answer: $\frac{1}{30}$

Q14 — $P(\text{between }17\text{ and }22)$

Find the probability (inclusive).

1. Numbers $17,18,19,20,21,22$ — $6$ cards.

Answer: $\frac{6}{30}=\frac{1}{5}$

Q15 — $P(\text{at least }20)$

Find the probability.

1. Numbers $20$ to $30$ — $11$ cards.

Answer: $\frac{11}{30}$

Q16 — $P(\text{not }27\text{ and not }29)$

Find the probability.

1. $2$ cards excluded out of $30$.

Answer: $1-\frac{2}{30}=\frac{14}{15}$

Q17 — $P(\text{not between }12\text{ and }15)$

Find the probability (excluding $12,13,14,15$).

1. $4$ cards excluded out of $30$.

Answer: $1-\frac{4}{30}=\frac{13}{15}$

Q18 — $P(\text{pass})=0.85$: find $P(\text{not pass})$

Find the probability Ayesha does not pass.

1. $P(\text{not pass})=1-0.85$.

Answer: $0.15$

Q19 — $P(\text{tail and dice}\ge 4)$

A coin and a die are used together. Find the probability.

1. $P(\text{tail})=\frac{1}{2}$, $P(\ge 4)=\frac{3}{6}=\frac{1}{2}$ — independent events.

Answer: $\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}$

Q20 — $P(\text{head and dice}=2\text{ or }3)$

Find the probability.

1. $P(\text{head})=\frac{1}{2}$, $P(2\text{ or }3)=\frac{2}{6}=\frac{1}{3}$.

Answer: $\frac{1}{2}\times \frac{1}{3}=\frac{1}{6}$

Q21 — $P(\text{head and tail on the same coin toss, and }6\text{ on dice})$

Find the probability.

1. A single coin toss cannot show both head and tail at once — impossible event.

Answer: $0$

Q22 — $P(\text{not tail and dice}=5)$

Find the probability.

1. "Not tail" means head: $P(\text{head})=\frac{1}{2}$, $P(5)=\frac{1}{6}$.

Answer: $\frac{1}{2}\times \frac{1}{6}=\frac{1}{12}$

Q23 — $P(\text{not head and dice}=5\text{ or }2)$

Find the probability.

1. "Not head" means tail: $P(\text{tail})=\frac{1}{2}$, $P(5\text{ or }2)=\frac{2}{6}=\frac{1}{3}$.

Answer: $\frac{1}{2}\times \frac{1}{3}=\frac{1}{6}$

Q24 — $P(\text{queen})$

From 52 cards, find $P(\text{queen})$.

1. $4$ queens out of $52$.

Answer: $\frac{4}{52}=\frac{1}{13}$

Q25 — $P(\text{neither queen nor jack})$

Find the probability.

1. $4$ queens $+4$ jacks $=8$ excluded cards.

Answer: $1-\frac{8}{52}=\frac{44}{52}=\frac{11}{13}$

Q26 — $P(\text{jack})$

Find the probability.

1. $4$ jacks out of $52$.

Answer: $\frac{4}{52}=\frac{1}{13}$

Q27 — $P(\text{no diamond})$

Find the probability.

1. $13$ diamonds out of $52$; $P(\text{no diamond})=1-\frac{13}{52}$.

Answer: $\frac{39}{52}=\frac{3}{4}$

Q28 — Horse-kick deaths: relative frequencies

$x=0,\dots ,6$; $f=60,50,87,40,32,15,10$ (total $294$). Find the relative frequency of each.

1. Divide each frequency by the total $294$.

Answer: $\approx 0.204,0.170,0.296,0.136,0.109,0.051,0.034$

Q29 — Defective products: relative frequencies

$0$–$8$ defectives; samples $=120,140,94,85,105,50,40,66,50$ (total $750$). Find each relative frequency.

1. Divide each by $750$.

Answer: $0.16,0.187,0.125,0.113,0.14,0.067,0.053,0.088,0.067$

Q30 — Quiz correct answers: relative frequencies

$X=0,\dots ,5$; $f=10,23,15,25,18,9$ (total $100$). Find each relative frequency.

1. Divide each by $100$.

Answer: $0.10,0.23,0.15,0.25,0.18,0.09$

Q31 — Percentage who like biryani

Biryani $=40$, out of $108$ students. Find the percentage.

1. $\frac{40}{108}\times 100$.

Answer: $\approx 37.04\%$

Q32 — Percentage who like chicken

Chicken $=21$, out of $108$. Find the percentage.

1. $\frac{21}{108}\times 100$.

Answer: $\approx 19.44\%$

Q33 — Least-liked food

Identify the least popular item.

1. Smallest count: Fresh Juice ($7$).

Answer: Fresh Juice

Q34 — Most-preferred food

Identify the most popular item.

1. Largest count: Biryani ($40$).

Answer: Biryani

Q35 — Expected frequency of sum $>8$ in $500$ trials (two dice)

Find the expected frequency.

1. Sum $>8$: sum $9$(4 ways), $10$(3), $11$(2), $12$(1) $=10$ outcomes out of $36$.
2. $P=\frac{10}{36}=\frac{5}{18}$; expected frequency $=500\times \frac{5}{18}$.

Answer: $\approx 138.9$

Q36 — Expectation: Rs.$120$ for at least $2$ heads in $3$ coin tosses

Find the expectation.

1. Sample space ($8$ outcomes); at least $2$ heads: HHH,HHT,HTH,THH $=4$ outcomes.
2. $P=\frac{4}{8}=\frac{1}{2}$; expectation $=\frac{1}{2}\times 120$.

Answer: Rs.$60$

Q37 — Expected frequencies for $200$ trials, given $P(x)$

$x=0,\dots ,6$; $P(x)=0.11,0.21,0.17,0.18,0.09,0.17,0.07$. Find each expected frequency.

1. Multiply each $P(x)$ by $200$.

Answer: $22,42,34,36,18,34,14$ (sum $=200$)

Q38 — Expected count of $5$ sixes in $200$ rolls, given $P=\frac{2}{5}$

Find the expected number of times.

1. Expected count $=200\times \frac{2}{5}$.

Answer: $80$

Q39 — $P(\text{green})$

Total balls $=23$. Find $P(\text{green})$.

1. $5$ green out of $23$.

Answer: $\frac{5}{23}$

Q40 — $P(\text{red})$

Find $P(\text{red})$.

1. $10$ red out of $23$.

Answer: $\frac{10}{23}$

Q41 — $P(\text{blue})$

Find $P(\text{blue})$.

1. $8$ blue out of $23$.

Answer: $\frac{8}{23}$

Q42 — $P(\text{not red})$

Find the probability.

1. $P(\text{not red})=1-\frac{10}{23}$.

Answer: $\frac{13}{23}$

Q43 — $P(\text{not green})$

Find the probability.

1. $P(\text{not green})=1-\frac{5}{23}$.

Answer: $\frac{18}{23}$

Q44 — $P(\text{exactly three heads})$

Sample space of $8$ outcomes. Find the probability.

1. Only HHH gives exactly $3$ heads.

Answer: $\frac{1}{8}$

Q45 — $P(\text{at least two tails})$

Find the probability.

1. TTT,TTH,THT,HTT — $4$ outcomes.

Answer: $\frac{4}{8}=\frac{1}{2}$

Q46 — $P(\text{not at least two heads})$

Find the probability.

1. "Not at least $2$ heads" means $0$ or $1$ head: TTT, and HTT/THT/TTH — $4$ outcomes.

Answer: $\frac{4}{8}=\frac{1}{2}$

Q47 — $P(\text{not exactly two heads})$

Find the probability.

1. Exactly $2$ heads: HHT,HTH,THH — $3$ outcomes; $P(\text{exactly }2)=\frac{3}{8}$.
2. $P(\text{not})=1-\frac{3}{8}$.

Answer: $\frac{5}{8}$

Q48 — $P(\text{king or jack of red colour})$

Find the probability.

1. Red king: $2$ (hearts, diamonds); red jack: $2$ — total $4$ cards.

Answer: $\frac{4}{52}=\frac{1}{13}$

Q49 — $P(\text{not "2" of club and spade})$

Find the probability.

1. Exactly $2$ specific cards excluded (2 of clubs, 2 of spades).

Answer: $1-\frac{2}{52}=\frac{25}{26}$

Q50 — Relative frequency of tails

No. of tails $=0,\dots ,6$; frequency $=110,90,105,80,76,123,16$ (total $600$). Find each relative frequency.

1. Divide each frequency by $600$.

Answer: $0.183,0.15,0.175,0.133,0.127,0.205,0.027$

Q51 — Relative and expected frequency of non-defective items

$25$ items, $8$ defective. Find the relative frequency of non-defective items and their expected frequency.

1. Non-defective $=25-8=17$; relative frequency $=\frac{17}{25}=0.68$.
2. Expected frequency (for the same lot of $25$) $=0.68\times 25$.

Answer: Relative frequency $=0.68$; expected frequency $=17$