Chapter 13 Probability
- If A and B are two events such that A ⊂ B and P(B) ≠ 0, then which of the following is true? A ⊂ B ⇒ A ∩ B = A ⇒ P(A ∩ B) = P(A) ∴ P(A/B) = P(A∩B)/P(B) = P(A)/P(B) But P(B) ≤ 1 ⇒ P(A/B) ≥ P(A). 1
- A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. Box contains 12 good and 3 bad oranges. If three oranges selected are good ones, the box is approved for sale. 3 good oranges may be selecte... 1
- A fair coin and an unbiased dice are tossed. Let A be the event ‘head appears on the coin’ and B be the event 3 on dice When a coin is thrown, head or tail will occur. ∴ Probability of getting head P(A) = 1/2 . when a dice is tossed 1, 2, 3, 4, 5, 6 one of the... 1
- A dice, marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event ‘number is even’ Even numbers on dice are 2, 4, 6. ∴ Probability of getting an even number = P(A) = 3/6 = 1/2 . There are two colours on the dice viz. red an... 1
- A dice is tossed thrice. Find the probability of getting an odd number at least once? If a dice is thrown and an odd number occurs, then it may result in 1, 3, 5. ∴ The probability of getting an odd number in a throw of a dice... 1
- Problem of solving specific problem independently by A and B are \(\frac{1}{2}\) and \(\frac{1}{3}\) respectively. Probability that A solves the problem = 1/2 ⇒ Probability that A does not solve the problem P( A ) = 1 – 1/2 = 1/2 . Probability that B solv... 1
- In a hostel, 60% of the students read Hindi newspaper, 40% read English newspaper and 20% read both Hindi (a) Let A be the event that a student reads Hindi newspaper, and B be the event that a student reads English newspaper. We are asked to find... 1
- The probability of obtaining an even prime number on each die, when a pair of dice is rolled is There are only one even prime number, i.e., 2, when a dice is rolled, the probability of getting an even prime number = 1/6 . When two dice... 1
- Two events A and B are said to be independent, if (A) A and B are mutually exclusive. (B) P(A’B’) (B) P(A’B’) = [1 – P(A)] [1 – P(B)] Two events are independent, if P(A ∩ B) = P(A) × P(B) or P(A’ ∩ B’) = P(A’) P(B’) = [1 – P(A)][1 – P(B)] 1
- Given that two numbers appearing on throwing two dice are different. Find the probability of the event When the numbers appearing on throwing two dice are different. ⇒ The numbers are not the same Total number of exhaustive cases = 36 Number o... 1
- Consider the experiment of throwing a dice, ‘if a multiple of 3 comes up, throw the dice again and if any other number The outcomes of the given experiment can be represented by the following tree diagram. The sample space of the experiment is, S=(1,H),(1,T),... 1
- If P(A) = \(\frac{1}{2}\), P(B) = 0, then P(A/B) is (A) 0 (B) \(\frac{1}{2}\) (C) not defined (D) 1 (C) not defined P(A) = 1/2 , P(B) = 0 P(A/B) = P(A∩B)/P(B) = P(A∩B)/0 = Not defined. 1
- If A and B are events such that P(A/B) = P(B/A), then (A) A ⊂ B (B) B = A (C) A ∩ B = ∅ (D) P(A) = P(B) (D) P(A) = P(B) is correct option : P(A/B) = P(B/A) ⇒ P(A∩B)/P(B) = P(A∩B)/P(A) ⇒ P(A) = P(B) 1
- If P(A) = \(\frac{6}{11}\), P(B) = \(\frac{5}{11}\) and P(A ∪B) = \(\frac{7}{11}\) P(A ∪ B) = P(A) + P(B) – P(A ∩ B) (i) P(A ∩ B) = P(A) + P(B) – P(A ∪ B) = 6/11 + 5/11 – 7/11 = 4/11 (ii) P(A/B)=P(A∩ B)/P(B))=4/11/5/11=4/5... 1
- Evaluate P(A ∪ B), if 2P(A) = P(B) = \(\frac{5}{13}\) and P(A/B) = \(\frac{2}{5}\) We have: 2P(A) = P(B) = 5/13 ⇒ P(A) = 5/26 , P(B) = 5/13 . So, P(A/B) = P(A∩B)/P(B) or 2/5 = P(A cap B)/5/13 ⇒ P(A ∩ B) = 2/5 × 5/13 = 2/13... 1
- If P(A) = 0.8, P(B) = 0.5 and P(B/A) = 0.4, find (i) P(A ∩ B) (ii) P(A/B) (iii) P(A ∪ B) (i) P(B/A) = P(A∩B)/P(B) ⇒ 0.4 = P(A∩B)/0.8 ∴ P(A ∩ B) = 0.4 × 0.8 = 0.32. (ii) P(A/B) = P(A∩B)/P(B) = 0.32/0.5 = 32/50 = 16/25 . (iii) P(A... 1
- Compute P(A/B), if P(B) = 0.5 and P(A ∩ B) = 0.32. P(A/B) = P(A∩B)/P(B) = 0.32/0.5 = 32/50 = 16/25 1
- Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.2, find P(E/F) and P(F/E). P(E/F) = E∩F/P(F) = 0.2/0.3 = 2/3 . P(F/E) = P(E∩F)/P(E) = 0.2/0.6 = 2/6 = 1/3 . 1