2 Marks Questions

Question 12 Marks

The probability of selecting a rotten apple randomly from a heap of $900$ apples is $0.18$ . What is the number of rotten apples in the heap?

Answer

Let $A$ be the event of selecting rotten apples.
Let $n$ be the number of rotten apples from the heap.
Probability of an event
$A=\frac{\text { Number of favourable outcomes }}{\text { Totalnumber of outcomes }}$
$P(A)=\frac{n}{900}$
$\Rightarrow 0.18=\frac{n}{900}$
$\Rightarrow n=162$
Thus, there are $162$ rotten apples in the heap.

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Question 22 Marks

Two different dice are tossed together. Find the probability
$(i)$ That the number on each die is even.
$(ii)$ That the sum of numbers appearing on the two dice is $5 .$

Answer

Two dice are tossed together, $n(S)=6^2=36$
(i) Let P be the event of getting an even number
$ n(P)=\left\{ (2,2),(2,4),(2,6),(4,2),(4,4),(4,6), (6,2),(6,4),(6,6) \right\}=9$
Probability that the number on each die is even
$=\frac{n(P)}{n(S)}=\frac{9}{36}=\frac{1}{4}$
$(ii)$ Let $Q$ be the event of getting a sum of $5$ on the two dice
$n(Q)=\{(1,4),(2,3),(3,2),(4,1)\}=4$
Probability that the sum of numbers appearing on the two dice is $=\frac{n(Q)}{n(S)}=\frac{4}{36}=\frac{1}{9}$

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Question 32 Marks

Rahim, tosses two different coins simultaneously. Find the probability of getting at least one tail.

Answer

Rahim tosses two coins simultaneously. The sample space of the experiment is $\{ HH , HT , TH$ and $TT\}$
Total number of outcomes $=4$
Number of outcomes which are in favour of getting at least one tail on tossing the two coins $=\{ HT , TH , TT \}$
Number of outcomes in favour of getting at least one tail $= 3$
Probability of getting at least one tail on tossing the two coins
$=\frac{\text { Number of favourable outcomes }}{\text { Total possible outcomes }}$
$=\frac{3}{4}$

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Question 42 Marks

A card is drawn at random from a well shuffled pack of 52 playing cards. Find the probability that the drawn card is neither a king nor a queen.

Answer

Total number of possible outcomes $=52$
Number of cards that are king or queen $=4+4=8$
$\therefore$ Number of other cards $=52-8=44$
Thus the number of cards which are neither a king nor a queen $=44$
Total number of favourable out comes $=44$
$\therefore$ Probability of getting a card which is neither a king nor a queen
$P(E)=\frac{\text { Number of favourable outcomes }}{\text { Total number of possible outcomes }}$
$
=\frac{44}{52}=\frac{11}{13}
$
Thus the probability of getting a card which is neither a king nor a queen is $\frac{11}{13}$

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Question 52 Marks

A card is drawn at random from a well-shuffled pack of cards. Find the probability of getting
(i) A red king
(ii) A queen or a jack

Answer

(i) In a deck of well-shuffled pack of 52 playing cards, there are 2 red cards with 'king' face cards. Therefore, the probability of getting a red king $=\frac{2}{52}=\frac{1}{26}$
(ii) In a deck of well-shuffled pack of 52 playing cards, there are 4'queen' cards and 4 'jack' cards. Therefore, the probability of getting a queen or a jack $=\frac{8}{52}=\frac{2}{13}$

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Question 62 Marks

A coin is tossed two times. Find the probability of getting at least one head.

Answer

When a coin is tossed two times, the elementary
outcomes are: $\{H H, H T, T H, T T\}$
Total outcomes $=4$
Favorable outcome for at least one head
$
=\{HT, TH, HH\}=3
$
Probability of getting at least one head
$
=\frac{\text { Favourable outcome }}{\text { Total outcome }}=\frac{3}{4}
$
Hence, the probability of getting at least one head is $\frac{3}{4}$.

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Question 72 Marks

A child has a dice whose six faces show the letters as shown below :

The die is thrown once. What is the probability of getting (i) A, (ii) D?

Answer

Total number of outcomes
$
=6(A, B, C, D, E, A)
$ (i) Number of favourable outcomes $( A )=2$
$
P(E)=\frac{2}{6}=\frac{1}{3}
$ (ii) Number of favourable outcomes (D) $=1$
$
P(E)=\frac{1}{6}
$

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Question 82 Marks

14 defective bulbs are accidentally mixed with 98 good ones. It is not possible to just look at the bulb and tell whether it is defective or not. One bulb is taken out at random from this lot. Determine the probability that the bulb taken out is a good one.

Answer

Total no. of outcomes $=98+14=112$
No. of favourable outcomes $=98$
(of good bulbs)
$
P(E)=\frac{98}{112}=\frac{49}{56}=\frac{7}{8}
$

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Question 92 Marks

A card is drawn at random from a pack of 52 playing cards. Find the probability of drawing a card which is neither a spade nor a king.

Answer

Number of possible outcomes $=52$
Number of favourable outcomes $=13+3=16$
$\therefore$ Probability (p) $=\frac{\text { favourable outcomes }}{\text { possible outcomes }}=\frac{16}{52}=\frac{4}{13}$
Hence, probability of card which is neither a spade nor a king $=\frac{4}{13}$

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Question 102 Marks

Three different coins are tossed simultaneously. Find the probability of getting exactly one head.

Answer

Number of possible out comes $=8$
$\{( H , H , H ),( H , H , T ),( H , T , H ),( T , H , H ),( H , T$, $T ),( T , H , T )( T , T , H ) \&(T, T , T )\}$
Number of favourable outcomes $=3$
$\{( H , T , T ),( T , H , T ),( T , T , H )\}$
$\therefore$ Probability $( p )=\frac{\text { favourable outcomes }}{\text { possible outcomes }}=\frac{3}{8}$
Hence, probability of getting exactly one head
$
=\frac{3}{8}
$

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