10th Maths 15.1
Chapter 15
Probability
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NCERT Class 10th solution of Exercise 14.1
Q1. Complete the following statements:
i) Probability of an event E + Probability of the event 'not E' =__________.
ii) The probability of an event that cannot happen is _____. Send an event is called __________.
iii) The probability of an event that is certain to happen is ______. Such an event is called_____.
iv) The sum of the probabilities of all the elementary events of an experiment is called________.
v) The probability of an event is greater than or equal to ______ and less than or equal to______.
Answer:
i) `1`
ii) `0`, impossible event
iii) `1`, sure or certain event
iv) `1`
v) `0, 1`
Q2. Which of the following experiments have equally likely outcomes? Explain.
i) A driver attempts to start a car. The car starts or does not start.
ii) A player attempts to shoot a basketball. She/he shoots or misses the shot.
iii) A trial is made to answer a true-false question. The answer is right or wrong.
iv) A baby is born. It is a boy or a girl.
Answer:
The experiments (iii) and (iv) have equally likely outcomes because they have equal possibilities of happening or not happening.
Q 3. Why is tossing a coin considered to be a fair way of deciding which team should get the ball at the beginning of a football game?
Answer:
When we toss a coin, the outcomes head and tail are equally likely. So, the result of an individual coin loss is completely unpredictable.
Q4. Which of the following cannot be the probability of an event?
A) `2/3` B) `-1.5` C) `15%` D) `0.7`
Answer:
B)
Q5. If `P(E) = 0.05,` what is the probability of not `E`?
Answer:
`P(overlineE) = 1 - P(E)` [not `E = P(overlineE)`]
`P(overlineE) = 1 - 0.05 = 0.95`
Q6. A bag contains lemon flavoured candies only. Malini takes out one candy without looking into the bag. What is the bag. What is the probability that she takes out
i) an orange flavoured candy?
ii) a lemon flavoured candy?
Answer:
i) `0` ii) `1`
Q7. It is given that a group of `3` students, the probability of `2` students not having the same birthday is `0.992.` What is the probability that the `2` students have the same birthday?
Sol. :
`P(E) = 1 - P(overlineE)`
`P(E) = 1 - 0.992`
`P(E) = 0.008`
Answer:
The probability that the `2` students have the same birthday is `0.008`.
Q8. A bag contains `3` red balls and `5` black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is (i) red? (ii) not red?
Sol. :
i) Total number of balls `= 3 + 5 = 8`
`n(S) = 8`
red balls `n(E) = 3`
`P(E) = (n(E))/(n(S))`
`P(E) = 3/8`
ii) not red balls `P(overlineE)`
`P(overlineE) = 1 - P(E)`
`P(overlineE) = 1 - 3/8 = 5/8`
Answer:
i) `3/8` ii) `5/8`
Q9. A box contains `5` red marbles, `8` white marbles, and `4` marbles. One marble is taken out of the box at random. What is the probability that the marble taken out will be (i) red? (ii) white? (iii) not green?
Sol. :
i) Total marbles `= 5 + 8 + 4 = 17`
`n(S) = 17`
red marbles `n(E_R) = 5`
`P(E_R) = (n(E_R))/(n(S))`
`P(E_R) = 5/17`
Answer:
`P(E_R) = 5/17`
ii) white marbles `n(E_W) = 8`
`P(E_W) = (n(E_W))/(n(S))`
`P(E_W) = 8/17`
Answer:
`P(E_W) = 8/17`
iii)
green marbles `n(E_G) = 4`
`P(E_G) = (n(E_G))/(n(S))`
`P(E_G) = 4/17`
not green `= 1 - P(E_G)`
`P(overline E) = 1 - 4/17 `
`P(overline E) = 13/17`
Answer:
`P(E_G) = 13/17`
Q10. A piggy bank contains hundred `50p` coins, fifty `₹ 1` coins, twenty `₹ 2` coins and ten `₹ 5` coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, what is the probability that the coin (i) will be a `50 P` coin? (ii) will not be a `₹ 5` coin?
Sol. :
Total coin `= 100 + 50 + 20 + 10 = 180`
`n(S) = 180`
i) `50 P` coin `= n(E) = 100`
`P(E) = (n(E))/(n(S))`
`P(E) = 100/180`
`P(E) = 10/18`
`P(E) = 5/9`
Answer:
`P(E) = 5/9`
ii) `₹ 5` coin `= n(E) = 10`
`P(E) = (n(E))/(n(S))`
`P(E) = 10/180 = 1/18`
not `₹ 5` coin ` = 1 - P(E)`
`P(overline E) = 1 - 1/18`
`P(overline E) = 17/18`
Answer:
not `₹ 5` coin ` = 17/18`
Q11. Gopi buys a fish from a shop for his aquarium. The shopkeeper takes out one fish at random from a tank containing `5` male fish and `8` female fish (see figure). What is the probability that the fish taken out is a male fish?
Sol. :
Total fishes `= 5 + 8 = 13`
`n(S) = 13`
Probability of Male `P(E)`
`P(E) = (n(E))/(n(S))`
`P(E) = 5/13`
Answer:
Probability of Male `= 5/13`
Q12. A game of chance consists of spinning an arrow which comes to rest pointing at one of the numbers `1, 2, 3, 4, 5, 6, 7, 8,` (see figure), and these are equally likely outcomes. What is the probability that will point at
i) `8`?
ii) an odd number?
iii) a number greater than `2`?
iv) a number less than `9`?
Sol. :
Total outcomes `1, 2, 3, 4, 5, 6, 7` and `8`
`n(S) =8`
i) `P(E) = (n(E))/(n(S))`
`P(E) = 1/8`
Answer:
`P(E) = 1/8`
ii) `E = 1, 3, 5, 7 = n(E) = 4`
`P(E) = (n(E))/(n(S))`
`P(E) = 4/8`
`P(E) = 1/2`
Answer:
P(E) = `1/2`
iii) `E = 3, 4, 5, 6, 7, 8 = n(E) = 6`
`P(E) = (n(E))/(n(S))`
`P(E) = 6/8`
`P(E) = 3/4`
Answer:
`P(E) = 3/4`
iv) `E = 1, 2, 3, 4, 5, 6, 7, 8 = n(E) = 8`
`P(E) =(n(E))/(n(S))`
`P(E) = 8/8 = 1`
Answer:
`P(E) = 1`
Q13. A die is thrown once. Find the probability of getting
i) a prime number,
ii) a number lying between `2` and `6`,
iii) an odd number.
Sol. :
Total number of outcome `n(S) = 6`
i) `E = 2, 3, 5 = n(E) = 3`
`P(E) = (n(E))/(n(S))`
`P(E) = 3/6 = 1/2`
Answer:
`P(E) = 1/2`
ii) `E = 2, 4, 5 = n(E) = 3`
`P(E) = (n(E))/(n(S))`
`P(E) = 3/6 = 1/2`
Answer:
`P(E) = 1/2`
iii) `E = 1, 3, 5 = n(E) = 3`
`P(E) = (n(E))/(n(S))`
`P(E) = 3/6 = 1/2`
Answer:
`P(E) = 1/2`
Q14. One card is drawn from a well-shuffled deck of `52` cards. Find the probability of getting
i) a king of red colour
ii) a face card
iii) a red face card
iv) the jack of hearts
v) a spade
vi) the queen of diamonds.
Sol. :
Total number of outcomes `n(S) = 52`
i) King of red colour `n(E) = 2`
`P(E) = (n(E))/(n(S))`
`P(E) = 2/52 = 1/26`
Answer:
`P(E) =1/26`
ii) Face card `n(E) = 12`
`P(E) = (n(E))/(n(S))`
`P(E) = 12/52 = 3/13`
Answer:
`P(E) = 3/13`
iii) Red Face card `n(E) = 6`
`P(E) = (n(E))/(n(S))`
`P(E) = 6/52 = 3/26`
Answer:
`P(E) = 3/26`
iv) Jack of Heart `n(E) = 1`
`P(E) = (n(E))/(n(S))`
`P(E) = 1/52`
Answer:
`P(E) = 1/52`
v) A Spades `n(E) = 13`
`P(E) = (n(E))/(n(S))`
`P(E) = 13/52 = 1/4`
Answer:
`P(E) = 1/4`
vi) Queen of Heart `n(E) = 1`
`P(E) = (n(E))/(n(S))`
`P(E) = 1/52`
Answer:
`P(E) = 1/52`
Q15. Five cards— the ten, jack, queen, king, and ace of diamonds, are well-shuffled with their face downwards. One card is then picked up at random.
i) What is the probability that the card is the queen?
ii) If the queen is drawn and put aside, what is the probability that the second card picked up is a) an ace? b) a queen?
Sol. :
Total Number of outcomes `n(S) = 5`
i) The number of queen `n(E) = 1`
`P(E) = (n(E))/(n(S))`
`P(E) = 1/5`
Answer:
`P(E) = 1/5`
Answer:
`P(E) = 1/5`
ii)
a) The number of Aces `n(E) = 1`
After the removal of the queen
Total number of outcomes `n(S) = 5 - 1=4`
`P(E) = (n(E))/(n(S))`
`P(E) = 1/4`
b) After the removal of the queens
Number of queen `n(E) = 0`
Total number of outcomes `n(S) = 4`
`P(E) = (n(E))/(n(S))`
`P(E) = 0/4 =0`
Answer:
a) `1/4` b) `0`
Q16. `12` defective pens are accidentally mixed with `132` good ones. It is not possible to just look at a pen and tell whether or not it is defective. One pen is taken out at random from this lot. Determine the probability that the pen is taken out is a good one.
Sol. :
Number of Defective pens `n(E_D) = 12`
Number of Good pens `n(E_G) = 132`
Total number of pens
`n(S) = n(E_D) + n(E_G)`
`n(S) = 12 + 132 = 144`
`P(E_G) = (n(E_G))/(n(S))`
`P(E_G) = 132/144 = 11/12`
Answer:
`P(E_G) = 11/12`
Q17. i) A lot of `20` bulbs contain `4` defective ones. One bulb is drawn at random from the lot. What is the probability that this bulb is defective?
ii) Suppose the bulb drawn in i) is not defective and is not replaced. Now one bulb is drawn at random from the rest. What is the probability that this bulb is not defective?
Sol. :
i) Number of bulbs `= 20`
`n(S) = 20`
Number of Defective bulbs `n(E_D) = 4`
`P(E_D) = (n(E_D))/(n(S))`
`P(E_D) = 4/20 = 1/5`
Answer:
`P(E_D) =1/5`
ii) After removal Defective bulb `n(S) = 20 - 1 = 19`
Number of Defective bulbs `= 4`
Number of Not Defective bulbs `= 19 - 4 = 15`
`n(E_(ND)) = 15`
`P(E_(ND)) = (n(E_(ND)))/(n(S))`
`P(E_(ND)) = 15/19`
Answer:
`P(E_(ND)) = 15/19`
Q18. A box contains `90` discs which are numbered from `1` to `90`. If one disc is drawn at random from the box, find the probability that it bears i) a two-digit number ii) a perfect square number iii) a number divisible by `5`.
Sol. :
Total possible outcomes `n(S) = 90`
i) Total numbers of two-digit no. `10, 11,..90` is `90 - 9 = 81`
`n(E) = 81`
`P(E) = (n(E))/(n(S))`
`P(E) = 81/90 = 9/10`
Answer:
`P(E) = 9/10`
ii) Total numbers of Perfect square no. `1, 4, 9...81` is `9`
`n(E) = 9`
`P(E) = (n(E))/(n(S))`
`P(E) = 9/90 = 1/10`
Answer:
`P(E) = 1/10`
iii) Total numbers divisible by `5` are `5, 10,... 90` i.e. `18`
`n(E) = 18`
`P(E) = (n(E))/(n(S))`
`P(E) = 18/90 = 1/5`
Answer:
`P(E) = 1/5`
Q19. A child has a die whose six faces show the letters as given below
[A] [B] [C] [D] [E] [A]
The die is thrown once. What is the probability of getting i) A? ii) D?
Sol. :
Total number of outcomes `n(S) = 6`
i) Probability of getting `A` is `n(E) = 2`
`P(E) = (n(E))/(n(S))`
`P(E) = 2/6 = 1/3`
Answer:
`P(E) = 1/3`
ii) Probability of getting `D` is `n(E) = 1`
`P(E) = (n(E))/(n(S))`
`P(E) = 1/6`
Answer:
`P(E) = 1/6`
Q20. Suppose you drop a die at random on the rectangular region shown in figure. What is the probability that it will land inside the circle with diameter `1 m`?
Area of rectangular region `= l times b`
`n(S) = 3 times 2`
`n(S) = 6 m^2`
Area of a circle `= pir^2`
`n(E) = pi times (1/2)^2`
`n(E) = pi times 1/4 = pi/4`
`P(E) = (n(E))/(n(S))`
`P(E) = (pi/4)/6`
`P(E) = pi/24`
Answer:
`P(E) = pi/24`
Q21. A lot consists of `144` ball pens of which `20` are defective and the others are good. Nuri will buy a pen if it is good, but will not buy if it is defective. The shopkeeper draws one pen at random and gives it to her. Where is the probability that
i) She will buy it?
ii) She will not buy it?
Sol. :
Total number ball pens `n(S) =144`
i) Good pen `=` total pen - defective pen
Good pen `= 144 - 20 = 124`
`n(E) = 124`
`P(E) = (n(E))/(n(S))`
`P(E) = 124/144 = 31/36`
Answer:
`P(E) = 31/36`
ii) not Good pen `n(overlineE) = 20`
`P(overlineE) = (n(overlineE))/(n(S))`
`P(overlineE) = 20/144 = 5/36`
Answer:
`P(overlineE) = 5/36`.
Q22. Refer if Example 13. i) Complete the following table:
ii) A student argues that there are `11` possible outcomes `2, 3, 4, 5, 6, 7, 8, 9, 10, 11` and `12.` Therefore, each of them has a probability of `1/11.` Do you agree with this argument? Justify your answer.
Sol. :
i)
Answer:
ii)
Answer:
No, The eleven sums are not equally likely.
Q23. A game consists of tossing a one rupee coin `3` times and noting its outcome each time. Hanif wins if all the tosses give the same result i.e., three heads or three tails, and loses otherwise. Calculate the probability that Hanif will lose the game.
Sol. :
Total possible outcomes `(HHH),``(HHT),``(HTH),``(THH),``(T``T``T),``(T``TH),``(THT),``(HT``T),`
`n(S) = 8`
Events for loses `(HHT),``(HTH),``(THH),``(HT``T),``(THT),``(T``TH)`
`n(E) = 6`
`P(E) = (n(E))/(n(S))`
`P(E) = 6/8 = 3/4`
Answer:
`P(E) = 3/4`
Q24. A die is thrown twice. What is the probability that
i) `5` will not come up either time?
ii) `5` will come up at least once?
[Hint: Throwing a die twice and throwing two dice simultaneously are treated as the same experiment ]
Sol. :
Total outcomes `n(S) = 36`
i) `5` will not come either time `n(E) =25`
`P(E) = (n(E))/(n(S))`
`P(E) = 25/36`
Answer:
`P(E) = 25/36`
ii) `5`will come at least once `n(E) =11`
`P(E) = (n(E))/(n(S))`
`P(E) = 11/36`
Answer:
`P(E) = 11/36`
Q25. Which of the following arguments are correct and which are not correct? Give reasons for your answer.
i) If two coins are tossed simultaneously there are three possible outcomes — two heads, two tails, or one of each. Therefore, for each of these outcomes, the probability is `1/3`
ii) If a die is thrown, there are two possible outcomes — an odd number or an even number. Therefore, the probability of getting an odd number is `1/2.`
Answer.
i) The given statement is Incorrect. We can classify the outcomes like this but they are not then 'equally likely'. The reason is that 'one of each' can result in two ways — from a head on the first coin and tail on the second coin or from a tail on the first coin and head on the second coin. This makes it twice as likely as two heads (or two tails).
ii) The given statement is Correct. The two outcomes considered in the question are equally likely.
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