# Theory of Expectation :: Problems on Tossing Coins : Probability Distribution

No. Problems & Solutions [Click Hide/Show to display the solutions below the question]
01. A person tosses a coin and is to receive Rs. 4 for a head and is to pay Rs. 2 for a tail. Find expectation and variance of his gains.

Solution
[Expectation: 1; Variance: 9]

02. Find the expected value of the number of tails appearing when two fair coins are tossed.

Solution
[Expectation: 1; Variance: 0.5]

03. An unbiased coin is tossed four times. If "x" denotes the number of heads, form the distribution of "x" by writing all the possible outcomes and hence calculate the expected value and variance of "x".

Solution
[Expectation: ; Variance: ]

04. A player tosses 3 fair coins. He wins Rs. 10 if 3 heads appear, Rs. 6 if 2 heads appear and Rs. 2 if 1 head appears on the other hand he losses Rs. 25 if 3 tails appear. Find the expected gain of the player.

Solution
[Expectation: ; Variance: ]

05. Throwing 2 unbiased coins simultaneously Mr. X bets with Mrs. X that he will receive Rs. 4 from her if he gets a head and he will give Rs. 4 to her otherwise. Find Mr. X’s expectation.

Solution
[Expectation: ; Variance: ]

06. A player tosses two fair coins. He wins Rs. 5 if 2 heads occur, Rs. 2 if 1 head occurs and Rs 1, if no head occurs. Find his expected gain
How much should he pay to play the game if it is to be fair?

Solution
[Expectation: ; Variance: ]

07. A player tosses three fair coins. He wins Rs. 12 if three tails occur, Rs. 7 if two tails occur and Rs. 2 if only one tail occurs. If the game is to be fair, how much should he wins or lose in case no tail occurs?

Solution
[Expectation: ; Variance: ]

08. Three coins, whose faces are marked 1 and 2, are tossed what is the expectation of total value of numbers on their faces?

Solution
[Expectation: ; Variance: ]

09. A weighted coin so that P(H) = 1/3 and P(T) = 2/3 is tossed until a head or 5 tails occur. Find the expected number of tosses of the coin.

Solution
[Expectation: ; Variance: ]

10. A person has 10 coins which he throws down in succession. He is to receive one rupee if the first falls head, another two rupees if the second also falls head, another four rupees if the third also falls head and so on, the amount doubling each time. But as soon as the coin falls tail he ceases to receive anything. What is the value of his expectation?.

Solution
[Expectation: ; Variance: ]

11. A coin is tossed until a head appears. What is expected number of tosses

Solution
[Expectation: ; Variance: ]

12. A, B,C in order toss a coin. The first one to throw a head wins a prize of Rs. 35, what are their respective mathematical expectations?

Solution
[Expectation: ; Variance: ]

13. Suppose in a game of coin tossing a person, say X will get Rs. 5 if the head turns up and will lose Rs. 4 if tail turns up. If n (number of trails) in which an unbiased coin is tossed are considered, what is the value of his expectation?

Solution
[Expectation: ; Variance: ]

No. Problems for Practice
01. A player tossed two coins if two heads show he wins Rs. 4 if one head shows he wins Rs. 2., but if two tails show he pays Rs. 3 as penalty. Calculate the expected value of the game to him.
02. A player tosses 3 coins. He wins Rs. 16 if 3 heads appear, Rs. 8 if 2 heads appear, Rs. 4 if 1 head appear and Rs. 2 if no head appears. Find his expected amount of winning.
03. A player tosses 3 fair coins. He wins Rs. 5 if 3 heads appear, Rs. 3 if 2 heads appear, Rs. 1 if 1 head occurs, one the other hand, he loses Rs. 15 if 3 tails occur. Find expected gain of the player
04. A person tosses two coins simultaneously and is to receive Rs.8 for two heads Rs.2 for one head and he is to pay Rs.6 for no head. Find his expectation.
05. A player tosses 3 fair coins. He wins Rs.8 if 3 heads occur Rs.3 if 2 heads occur, Re.1 if only 1 head occurs. If the game is to be fair how much would he lose if no head occurs?
06. A balanced coins is tossed 4 times. Find the probability distribution of the number of heads and its expectation.
07. A coin is tossed until a head or 5 tails occur. Find the expected number of tosses of the coin