An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards numbered 1, 2, 3, ..., 9 is randomly picked and the number on the card is noted. The probability that the noted number is either 7 or 8 is:
Answers
Answer:
1) An unbiased coin is tossed.
Total number of possible outcomes = 2 (head,tail).
The probability of getting a Head or a Tail is the same.
The probability that the head is the outcome P(H) = 1/2
The probability that the tail is the outcome P(T) = 1/2 [ Since P(H) + P(T) = 1]
2) If, the outcome is a head:
Given that; a pair of dice is rolled
Total possible outcomes when a pair of dice is rolled = 36
Which are ;
{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),
(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}
Probability of getting a sum of 7, P(S1) = Number of occurrences with a sum of 7 / Total number of occurrences
= 6/36
Since, possible outcomes are ; {(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)} = 6
Probability of getting a sum of 8, P(S2) = Number of occurrences with a sum of 8 / Total number of occurrences
= 5/36
Since, possible outcomes are ; {(2,6),(3,5),(4,4),(5,3),(6,2)} = 5
So, the probability of getting a sum of 7 or 8 P(H1), when the unbiased coin tossed shows Head
P(H1) = P(H).P(S1) + P(H).P(S2)
= 1/2 . 6/36 + 1/2 . 5/36
= 11/72
3) If, the outcome is a tail:
Given that; a card is picked from a pack of cards 1,2,3,4,….9
Total possible outcomes = 9
{1,2,3,4,5,6,7,8,9}
Probability of picking the number 7, P(C1) = Number of occurrences with the number 7 / Total number of occurrences
= 1/9
Probability of picking the number 8, P(C2) = Number of occurrences with the number 8 / Total number of occurrences
= 1/9
So, the probability of getting the numbers 7 or 8 P(T1), when the unbiased coin tossed shows Tail
P(T1) = P(T).P(C1) + P(T).P(C2)
= 1/2 . 1/9 + 1/2 . 1/9
= 2/18
4) The probability that the required number noted is either 7 or 8 is:
Since Point (2) and Point (3) are mutually exclusive events;
The required probability is;
P(H1) + P(T1)
= 11/72 + 2/18 = (11+8)/72 = 19/72 = 0.264