Dear Reader, Below are three probability questions to practice. First question is straight forward while second and third bear little twists.
Question 1
For the IPL 5 cricket matches Mr. Bala Josiar has been predicting the winner of each match with amazing success. It is rumored that in a match between 2 teams X and Y, Bala Josiar picks X with the same probability as X's chances of winning. Let's assume such rumors to be true and that in a match between Rajasthan Royals and Chennai Super Kings, Chennai Super Kings the stronger team seems to have a probability of 3/4 of winning the game. What is the probability that Bala Josiar will correctly pick the winner of the Rajasthan Royals–Chennai Super Kings game?
a) 7/ 16 b) 9/16 c) 3/4 d) 10/16
Answer : d) 10/16
Solution :
Probability that CSK wins = 3/4
Probability that RR wins = 1 - Probability that CSK wins = 1 - 3/4 = 1/4
Probability of picking a winner = Probability of picking RR x Probability that RR wins + Probability of picking CSK x Probability that CSK wins
Bala Josiar picks with the same probability of a team’s chance of winning. This means, Probability of picking RR = Probability that RR wins and Probability of picking CSK = Probability that CSK wins
Therefore, Probability of picking a winner = (Probability that CSK wins)2 + (Probability that RR wins)2
= ( 3/4 )2 + (1/4)2
= 9/16 + 1/16
= (9 + 1) / 16
= 10/16
Question 2
Mr. Randor Guy, the famous astrologer staying at Anna Nagar, Chennai has been able to predict the winner of each Hockey match with amazing success. It is told in the sports circle that in a match between two teams A and B, Mr. Randor Guy picks A team with the same probability as A’s chances of winning. But he seems to be incapable of predicting a tie. Let us assume such rumors to be true and that in a match between Pakistan and India, India the stronger team has a probability of 2/3 of winning the game. Based on past data it is also estimated that the probability of match getting tied is 1/9. What is the probability that Randor Guy will correctly pick the winner of the Pakistan-India game?
a) 4/9 b) 2/3 c) 52/81 d) none of these.
Answer : c) 52/81
Solution :
This question is very similar to the previous question, except the fact that there is a probability of tie as well.
Probability of India winning = 2/3
Probability of Pakistan winning = 1 - Probability of India winning - Probability of Tie
= 1 - 2/3 - 1/9 = 1 - 5/9 = 4/9
So probability of picking a winner = probability of picking Pakistan* Probability of Pakistan winning + probability of picking India* Probability of India winning + probability of picking a tie x probability of tie ....(1)
Randor Guy picks with the same probability of a team’s chance of winning.
Therefore, Probability of picking Pakistan = Probability of Pakistan winning, Probability of picking India = Probability of India winning. Also we know that he is unable to predict a tie. Therefore, probability of picking a Tie = 0.
Applying above in eq (1) we get,
probability of picking a winner = (Probability of Pakistan winning )2+(Probability of India winning)2+ 0 x probability of tie
= ( 4/9 * 4/9) + ( 2/3* 2/3) + 0
= 16/81 + 4/9
= (16 + 36) / 81
= 52/81
Question 3
Om Navasakthi Astrologers , Chennai has been able to predict the winner of each football match with amazing success. It is told in the sports circle that in a match between two teams A and B, Om Navasakthi Astrologers picks A team with the same probability as B’s chances of winning and vice versa. Let us assume such rumors to be true and that in a match between Germany and Brazil, Brazil the stronger team has a probability of 4/7 of winning the game. What is the probability that Om Navasakthi Astrologers will correctly pick the winner of the Germany-Brazil game?
a) 24/49 b) 4/7 c) 18/49 d) none of these.
Answer : a) 24/49
This is similar to earlier questions except the fact that Astrologers pick A team with the same probability as B’s chances of winning and pick B team with the same probability as A's chances of winning.
Probability of Brazil winning = 4/7.
Therefore, Probability of Germany winning = 1 - 4/7 = 3/7
So probability of picking a winner = probability of picking Brazil*Probability of Brazil winning + probability of picking Germany * Probability of Germany winning ....(1)
Astrologers pick A team with the same probability as B’s chances of winning and pick B team with the same probability as A's chances of winning.
Therefore, Probability of Brazil winning = probability of picking Germany and Probability of Germany winning = probability of picking Brazil
Applying this in eq (1) we get
probability of picking a winner = ( 3/7 * 4/7) + ( 4/7* 3/7)
= 12/49 + 12/49
= 24/49