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Q25(i):
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 $\frac{1}{3}$.
Solution :
Given: Two coins are tossed simultaneously. The argument states that there are three possible outcomes: two heads (HH), two tails (TT), or one of each (HT or TH). Based on this, it claims the probability of each outcome is $\frac{1}{3}$.
To Find: Determine if the argument is correct or incorrect and provide a mathematical justification.
Step 1: Identifying the Sample Space
When two fair coins are tossed simultaneously, each coin has two possible outcomes: Head ($H$) or Tail ($T$). Let the outcomes of the first coin be $C_1$ and the second coin be $C_2$. The total number of possible outcomes is determined by the Fundamental Counting Principle: $2 \times 2 = 4$.
The sample space $S$ is the set of all possible elementary events:
$S = \{HH, HT, TH, TT\}$
Where:
$HH$ = Head on both coins
$HT$ = Head on the first coin, Tail on the second
$TH$ = Tail on the first coin, Head on the second
$TT$ = Tail on both coins
Step 2: Analyzing the Argument's Outcomes
The argument suggests three outcomes: "two heads", "two tails", or "one of each".
Let us map these to our sample space $S$:
- Two heads: $\{HH\}$ (1 outcome)
- Two tails: $\{TT\}$ (1 outcome)
- One of each: $\{HT, TH\}$ (2 outcomes)
Step 3: Calculating Probabilities
Using the classical definition of probability, $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$.
The total number of outcomes in the sample space is $n(S) = 4$.
Calculating the probability for each category defined in the argument:
1. Probability of two heads ($P_{HH}$):
$P(HH) = \frac{1}{4}$
2. Probability of two tails ($P_{TT}$):
$P(TT) = \frac{1}{4}$
3. Probability of one of each ($P_{one\_of\_each}$):
$P(HT \text{ or } TH) = \frac{2}{4} = \frac{1}{2}$
Step 4: Comparison and Conclusion
The argument claims that the probability for each of the three outcomes is $\frac{1}{3}$. However, our calculations show:
$P(HH) = \frac{1}{4} \neq \frac{1}{3}$
$P(TT) = \frac{1}{4} \neq \frac{1}{3}$
$P(one\_of\_each) = \frac{1}{2} \neq \frac{1}{3}$
The error in the argument lies in the assumption that the three outcomes are "equally likely". In probability theory, the classical formula $P(E) = \frac{1}{n}$ only applies when all elementary events in the sample space are equally likely. Since the event "one of each" consists of two distinct elementary outcomes ($HT$ and $TH$), it is twice as likely to occur as "two heads" or "two tails".
Final Answer: The argument is incorrect. The outcomes are not equally likely; the probability of getting two heads is $\frac{1}{4}$, two tails is $\frac{1}{4}$, and one of each is $\frac{1}{2}$.
More Questions from Class 10 Mathematics Probability EXERCISE 14.1
- Q1(i): Complete the following statements: (i) Probability of an event $E$ + Probability of the event ‘not $E$’ = .
- Q1(ii): Complete the following statements: (ii) The probability of an event that cannot happen is . Such an event is called .
- Q1(iii): Complete the following statements: (iii) The probability of an event that is certain to happen is . Such an event is called .
- Q1(iv): Complete the following statements: (iv) The sum of the probabilities of all the elementary events of an experiment is .
- Q1(v): Complete the following statements: (v) The probability of an event is greater than or equal to and less than or equal to .
- Q2(i): 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.
- Q2(ii): Which of the following experiments have equally likely outcomes? Explain. (ii) A player attempts to shoot a basketball. She/he shoots or misses the shot.
- Q2(iii): Which of the following experiments have equally likely outcomes? Explain. (iii) A trial is made to answer a true-false question. The answer is right or wrong.
- Q2(iv): Which of the following experiments have equally likely outcomes? Explain. (iv) A baby is born. It is a boy or a girl.
- Q3: 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?
- Q4: Which of the following cannot be the probability of an event?
- Q5: If $P(E) = 0.05$, what is the probability of ‘not $E$’?
- Q6(i): A bag contains lemon flavoured candies only. Malini takes out one candy without looking into the bag. What is the probability that she takes out (i) an orange flavoured candy?
- Q6(ii): A bag contains lemon flavoured candies only. Malini takes out one candy without looking into the bag. What is the probability that she takes out (ii) a lemon flavoured candy?
- Q7: It is given that in 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?
- Q8(i): 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 ?
- Q8(ii): 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 (ii) not red?
- Q9(i): A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out of the box at random. What is the probability that the marble taken out will be (i) red ?
- Q9(ii): A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out of the box at random. What is the probability that the marble taken out will be (ii) white ?
- Q9(iii): A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out of the box at random. What is the probability that the marble taken out will be (iii) not green?
- Q10(i): 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 ?
- Q10(ii): 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 (ii) will not be a ` 5 coin?
- 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 Fig. 14.4). What is the probability that the fish taken out is a male fish?
- Q12(i): 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 Fig. 14.5 ), and these are equally likely outcomes. What is the probability that it will point at (i) 8 ?
- Q12(ii): 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 Fig. 14.5 ), and these are equally likely outcomes. What is the probability that it will point at (ii) an odd number?
- Q12(iii): 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 Fig. 14.5 ), and these are equally likely outcomes. What is the probability that it will point at (iii) a number greater than 2?
- Q12(iv): 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 Fig. 14.5 ), and these are equally likely outcomes. What is the probability that it will point at (iv) a number less than 9?
- Q13(i): A die is thrown once. Find the probability of getting (i) a prime number;
- Q13(ii): A die is thrown once. Find the probability of getting (ii) a number lying between 2 and 6;
- Q13(iii): A die is thrown once. Find the probability of getting (iii) an odd number.
- Q14(i): One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting (i) a king of red colour
- Q14(ii): One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting (ii) a face card
- Q14(iii): One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting (iii) a red face card
- Q14(iv): One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting (iv) the jack of hearts
- Q14(v): One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting (v) a spade
- Q14(vi): One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting (vi) the queen of diamonds
- Q15(i): 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?
- Q15(ii)(a): 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. (ii) If the queen is drawn and put aside, what is the probability that the second card picked up is (a) an ace?
- Q15(ii)(b): 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. (ii) If the queen is drawn and put aside, what is the probability that the second card picked up is (b) a queen?
- 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 taken out is a good one.
- Q17(i): (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?
- Q17(ii): (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 ?
- Q18(i): 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
- Q18(ii): 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 (ii) a perfect square number
- Q18(iii): 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 (iii) a number divisible by 5.
- Q19(i): A child has a die whose six faces show the letters as given below: The die is thrown once. What is the probability of getting (i) A?
- Q19(ii): A child has a die whose six faces show the letters as given below: The die is thrown once. What is the probability of getting (ii) D?
- Q20: Suppose you drop a die at random on the rectangular region shown in Fig. 14.6. What is the probability that it will land inside the circle with diameter $1$m?
- Q21(i): 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. What is the probability that (i) She will buy it ?
- Q21(ii): 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. What is the probability that (ii) She will not buy it ?
- Q22(i): Refer to Example 13. (i) Complete the following table:
- Q22(ii): Refer to Example 13. (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 $\frac{1}{11}$. Do you agree with this argument? Justify your answer.
- 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.
- Q24(i): A die is thrown twice. What is the probability that (i) 5 will not come up either time? [Hint : Throwing a die twice and throwing two dice simultaneously are treated as the same experiment]
- Q24(ii): A die is thrown twice. What is the probability that (ii) 5 will come up at least once? [Hint : Throwing a die twice and throwing two dice simultaneously are treated as the same experiment]
- Q25(ii): Which of the following arguments are correct and which are not correct? Give reasons for your answer. (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 $\frac{1}{2}$.
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