[확률과 랜덤변수] Conditional Probability를 가지고 있는 두개의 동전 던지기의 분석

소개

두개의 동전을 무한번 던질때 얻을수 있는 확률에 대해 수학적으로 분석한 것과 매틀랩의 이용하여 분석한 것과 일치하는지 여부를 알아보는 프로젝트입니다.
Fair한 동전과 Unfair한 동전 두개가 있는데 던져서 앞면이 나오면 Fair한 동전을 던지고 뒷면이 나오면 Unfair한 동전을 던지는 조건을 가지고 있습니다.

영어로 작성되었으며 매틀랩코드도 다 포함되어 있습니다.

목차

Tossing Two Coins
Problem 1) Write a prgram to simulate the above coin-tossing experiment.
Problem 2) Plot the relative frequency of tossing Coin0 as a function of the number of trials n. You need to repeat the experiment sufficiently many times such that the relative frequency converges.
Problem 3) Can you find analytically the probability of choosing Coin0 for tossing as the number of trials goes to infinity? If so, what is it?
Problem 4) Does your analysis in p3) agree with simulation result in p2)?

본문내용

Tossing Two Coins
Consider tossing two coins repeatedly. One (Coin 0) is fair and the other(Coin 1) is biased. The probability of head for the biased coin (Coin 1) is1/4. After tossing a coin, we choose Coin 0 if the outcome is head and Coin 1 if the outcome is tail for the next toss. Assume that we toss Coin 0 for the first trial.

Problem 1) Write a prgram to simulate the above coin-tossing experiment.

First, let analysis the above experiment. What kind of coin will be tossed is decided by previous tossing. We choose Coin 0 if the outcome of previous tossing is head and Coin 1 if the outcome of previous tossing is tail for the next toss.
Our team embodied and simulated the above coin-tossing experiment in Matlab. Because we don`t programing a practical coin-tossing, we mapped probabilities of head and tail for a pair coin and a biased coin by rand function. In case of coin0, if a result of rand function is greater than 0.5, mapping Head. If a result of rand function is less than 0.5, mapping Tail. In case of coin1, if a result of rand function is greater than 0.75, mapping Head. If a result of rand function is less than 0.75, mapping Tail.
Our focus of this experiment is number of coin 0 for n times trials. Therefore, we must observe what kind of coin is tossed and count whenever tossing a coin.
In conclusion, if present_coin=0 & a result of rand >= 0.5 or present_coin=1 & a result of rand >= 0.75, we must count coin0. (because outcomes of above two case is head, we choose coin0 in the next trial.) In opposite case, we count coin1. (The number of coin1 is used only at the time of checking. If Coin0+Coin1=n, the simulating program is correct.)

참고자료

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