Suppose I have 6 coins. Two are rigged to end up heads 80% of the time when flip

Question

Suppose I have 6 coins. Two are rigged to end up heads 80% of the time when flipped, and we call these "type A" coins, one is a fair coin (shows heads 50% of the time), and we call this a "type B" coin, and three are rigged to show up heads 25% of the time, and we call these "type C" coins. The 6 coins are put into a bag. We now randomly draw one of the coins from the bag, and record a sequence of flips of that coin. Suppose we perform 3 successive flips. What is a mathematical description of the event space sigma we are working with, and the assumptions on your probability model. Does sigma have a uniform probability measure on it? Letting A, B, and C denote the event that we selected that particular type coin from the bag, find P(A), P(B), P(C). Suppose we flip the coin once. Call this result f_1. Find the probability that this flip results in a heads, i. e., find P(f_1 = H). Give your answer as a fraction in lowest terms. Now find P(A|f_1 = H), P(B|f_1= H), and P(C|f_1 = H). Give your answers as fractions in lowest terms. Now we flip the coin one more time, calling this result f_2. This second flip results in a tails. Find the probability P((f_1, f_2) = (H, T)). Give your answer as a fraction in lowest terms. Now find P(A| (f_1, f_2) = (H, T)), P(B| (f_1, f_2) = (H, T)), and P(C| (f_1, f_2) = (H, T)). Give your answers as fractions in lowest terms. Now we flip the coin for the third time, calling the result f_3. This third flip results in a heads. Find the probability P((f_1, f_2, f_3) = (H, T, H)). Give your answer as a fraction in lowest terms. Finally find the conditional probabilities that you have a type A, B, or C coin given the sequence of flips (f_1, f_2, f_3) = (H, T, H). Give your answers as fractions in lowest terms. Rank your confidence in which type coin you have.

Explanation / Answer

Probability means the chance or the likelihood of
occurrence of an event. It is a numerical value that is lying between 0 and 1. When the event is impossible probability of occurring that event is 0, when the event is sure, probability of occurring that event is 1 and all other events will have probability between 0 and 1.

There are mainly 3 definitions of probability

Theoretical or classical definition of probability

Let a random experiment produce only a finite number of mutually exclusive and equally likely outcomes. Then the probability of an event A is defined as
                P(A) = Number of favorable outcomes to A/ Total number of outcomes
Theoretical probability is also known as Classical or A Priori probability. Classical probability is also known as statistical probability.
Example: -
Consider the experiment of tossing a coin.What is the probability of getting a tail?
Solution: -
We know that the probability of an event A is given by
P(A) = Number of favorable outcomes to A/ Total number of outcomes
The Sample space of the experiment of tossing a coin is given by,
S = {Head, Tail}
Number of favorable outcomes to tail = 1
Total number of outcomes = 2
So P(getting a tail) = 1/2

Empirical definition of probability

Let A be an event of a random experiment.
Let the experiment be repeated n number of times out of which A occurs f
times. Then f/n is called frequency ratio. The limiting value of the
frequency ratio as the number of repetitions becomes infinitely large is called
probability of the event A.

It is sometimes known as experimental probability also.

Example: -
The following below shows the number of heads appearing when 8 coins are tossed.
x;         0          1          2          3          4          5          6          7          8
f:          1          9          26        59        72        52        26        7          1
Find the probability of getting heads
(i) less than 4   
(ii)equal to 5  
(iii) more than 6
Solution: -
Here total number of heads = 1 + 9 + 26 + 59 + 72 + 52 + 26 + 7 + 1 = 253
(i) We have to find the probability of getting heads less than 4.
There are 1 + 9 + 26 + 59 = 95 cases in which there are less than 4 heads.
Therefore P(heads less than 4) = 95/253 = 0.37
(ii) We have to find the probability of getting heads equal to 5.
There are 52 cases in which there are 5 heads.
Therefore P(heads equal to 5) = 52/253 = 0.21
(iii) We have to find the probability of getting more than 6 heads.
There are 7 + 1 = 8 cases in which there are more than 6 heads.
Therefore P(heads more than 6) = 8/253 = 0.03

Axiomatic definition of probability

Let S be the sample space of a random experiment.
Let A be an event of a random experiment.
According to this definition Probability of an event is a real number which
satisfies 3 axioms.
(i) Axiom 1:P(A) 0
(ii) Axiom 2 :P(S) = 1
(iii) Axiom 3: P(AB) = P(A) + P(B) where A and B are disjoint events
Therefore P(A) is a real valued, non negative, totally additive set function. Therefore P(A) is a measure and it is the probability of A.
Axiomatic definition is usually not used to work out problems

Properties of Probability

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.