Question

Your friend has a biased coin at his home and chalanges you to determine its bias. That is the probability of heads.You both have a personal communication system which can transmit / receive bits. You ask your friend to toss the coin 10000 times, send a 0 when a tail comes up and a 5when a head comes up.He/She agrees because whatever you receive it is any way corrupted by noise. Which has a Gaussian PDF with mean µ and variance σ^2. Since you taken this course and can use some tricks to estimate the noise parameters. You put an additional request to your friend and ask him/her to simply send 1000 zeros before sending the coin toss results.Using these 11000 samples of data characterize the noise and then find the bias of the coin.

Answer 1

Answer:

We know foo() returns 0 with 60% probability. How can we ensure that 0 and 1 are returned with a 50% probability?

The solution is similar to this post. If we can somehow get two cases with equal probability, then we are done. We call foo() two times. Both calls will return 0 with a 60% probability. So the two pairs (0, 1) and (1, 0) will be generated with equal probability from two calls of foo(). Let us see how.

(0, 1): The probability to get 0 followed by 1 from two calls of foo() = 0.6 * 0.4 = 0.24

(1, 0): The probability to get 1 followed by 0 from two calls of foo() = 0.4 * 0.6 = 0.24

So the two cases appear with equal probability. The idea is to return consider only the above two cases, return 0 in one case, return 1 in other case. For other cases [(0, 0) and (1, 1)], recur until you end up in any of the above two cases.

Answer 2

Concept:

A continuous probability distribution for a real-valued random variable is known as a normal distribution in statistics. It is sometimes referred to as a Gaussian, Gauss, or Laplace-Gauss distribution.

So the formula to find the probability of biased coins,

P(A|B)=P(B|A) P(A)P(B)

You're interested in knowing the likelihood of P(biased coin | three heads).

For example:

1.The likelihood of three heads on a fair coin is 0.53=1/8.

2.P(biased coin)=1/100 is the probability of selecting the biased coin.

Given:

Success of probability (p) = 0.4

Failure of probability (q) = (1 - 0.4) = 0.6

Number of trials per biased coin (n) = 10000

Find:

We have to find the standard deviation.

Solution:

We will be done if we can find two situations with an equal chance. We make two calls to foo(). In 60% of the cases, both calls will result in a return value of 0. Therefore, after two calls of foo, the two pairs (0, 1) and (1, 0) will be created with equal chance (). Let's examine how.

Variance = n x p x q

Variance = 10000 x 0.4 x 0.6

Variance = 2400

Now, standard deviation = 2400

Standard deviation = 48.98

Hence, the variance of the biased coin is 2400 and standard deviation is 48.98.

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