L03.6 Independence Versus Conditional Independence

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MIT RES.6-012 Introduction to Probability, Spring 2018
Instructor: John Tsitsiklis

License: Creative Commons BY-NC-SA

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We can easily calc the following probabilities from the prob. tree graph. Just calc the product of the prob of your choice of path

= 0.9^11
= 0.1^11

P(H1, H2....H11) = + = 0.5 * [0.9^11+0.1^11]

similarly P(H1, H2....H10) = 0.5 * [ 0.9^10 + 0.1^10 ]

P(H11|H1, H2....H10) = [ P(H1, H2...H10) and P(H11) ] / P(H1, H2...H10)

= P(H1, H2...H11) / P(H1, H2..H10)
= 0.5 * [0.9^11+0.1^11] / 0.5 * [ 0.9^10 + 0.1^10 ]
= [0.9^11+0.1^11] / [ 0.9^10 + 0.1^10 ]

As we exponentiate smaller decimal value, the resulting values tend to be nearer to 0, So 0.1 ^ 11 and 0.1 ^ 10 will be near 0

so P(H11|H1, H2, ....H10) ~ 0.9 ^11 / 0.9 ^ 10 ~ 0.9

Since P(H11) = 0.5 (from the video)

P(H11) !=

So that means even though the successive coin tosses are conditionally independent on the choice of coin, the probabilistic model taken as whole doesn't guarantee independence

abir

From my view the second probability is (0.9^11+0.1^11)/(0.9^10+0.1^10), which approximates 0.9.

vincentschwartz

How to apply Bayes rule for the second probability? P(toss 11 = H | first 10 tosses are heads) = P(toss 11 = H (intersection) first 10 tosses are heads)/P(first 10 tosses are heads). The denominator here is 0.5. What about numerator?

ks

Sir can we also say since probability of choosing coin a and b are disjoint, so they are not independent

rahularanger

Can anybody clear my confusion. Why he takes coin A in second probability(H11|A) instead of B And also tell me how can be the answer is 0.9

SyedaHamdani-hj

i am so sorry to say that, this is the fucking real the fucking knowledge, you are the best teacher God your mouth never stop saying the good things like you are singing a beautiful songs

gghhff

First episode i didn't like. Professor just went ahead and guessed the last part instead of using the numbers to further explain the comparison of the 2 probabilities

Younessss_

i am so sorry to say that, this is the fucking real the fucking knowledge, you are the best teacher God your mouth never stop saying the good things like you are singing a beautiful songs

gghhff

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