Dependent probability
Example: Bag of unfair coins Example where the probability of an outcome is dependent on which coin you happen to pick
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- Lets do another one of these dependent probability problems
- You have 4 coins in a bag
- 3 of them are unfair in that they have a 45% chance of
- coming up tails when flipped
- the rest are fair
- so for the rest of them they have a 50% chance of tails or 50% chance of coming heads
- You randomly choose one coin from the bag and flip it 4 times
- What is the percent probability of coming 4 heads
- So lets think about it
- when we put our hand in the bag and and we take one of the coins out
- there is some probability that we get an unfair coin
- and 3 of the 4 coins are unfair so there is a 3/4 probability that we get an unfair coin
- and there is only one of the 4 coins which is fair so there is a 1/4 probability
- that i get a fair coin
- now, given that I have so unfair-- Lets remind ourselves
- a unfair coin has a 45% chance of coming up tails so this means that I have a 45% chance
- of tails which also means -- we have to be careful here
- because they are asking us about heads
- If I have a 45% chance of getting tails that means
- I have a 55% chance of getting heads, right?
- when I have a 100% chance of getting on of these 2
- If its 45% for tails then 100 - 45 is 55 for heads
- For the fair coin I have a 50% chance of tails and 50% for heads
- fair enough
- now I want to know in either of these senarios
- what is the percent probability of getting 4 heads
- so if given--I have got the unfair coin the probability of getting
- 4 heads is going to be 55% for each of those flips
- so the probability of getting exactly 4 heads is going to be
- 0.55 times 0.55 times 0.55 times 0.55 and so the probability
- of picking an unfair coin and getting 4 heads in a row
- is going to be equal 3/4 times all of this business over here
- so thats 3/4 times and this is 0.55 times itself times 4 times
- so I could write it as 0.55 to the 4th power
- And we will get the calculator out in a second
- and actually calculate what this is
- now lets do the same thing for the fair coin
- If I did pick a fair coin
- the probability of getting heads four times in a row
- is going to be 0.5 times 0.5 times 0.5 times 0.5
- or the probability of getting the fair coin which is 1/4 chance and getting 4 heads in the row is going to be
- 1/4 times all of this so its going to be 1/4 times -- this is just 0.5 times itself 4 times so that's
- so thats 0.5 to the 4th power
- so lets get the calculator out to caculate one of these
- so we get 3 divided by 4 times -- and it knows when I do the multiplication
- its not in the denominator here
- so its 3/4 times -- and I will just do it in parentheces
- not to do in parenthesies because it the order of operation
- so .55 to the fourth power is equal to 0. -- let me write it down
- take it off the screen so that I can write properly
- actually let me do both of these calculations
- so this probability is that one right over there
- and then this one down here is 1 divided by 4 times .5 to the 4th power
- so its equal to that right over there
- and so lets be clear
- the probability of picking the unfair coin and then getting 4 heads in a row
- is this top number its like roughly 6.9% chance
- that you get the unfair coin and then get 4 heads in a row
- the probability that you get the fair coin and then get 4 heads in a row
- is even lower -- its only a 1.6% chance
- now the probability of getting 4 heads in a row either way is going to be
- the sum of this and this or the sum of that and that
- which is going to be --let me get my calculator out
- so its going to be equal to I could just take the previous Ans
- let me just e type it so that I don't confuse you
- so .015625 + .0686296875 -- I am going to round it anyway
- wont matter too much
- so if I take the sum -- let me take this of screen so I can still see
- and that let me write it
- so what I got here this one is 0.068629 -- I willl round it 7
- and this down here was 0.015625
- and when you add these 2 up
- because we just care about getting 4 heads either way
- there is a probability of getting it this way, the unfair coin
- this is the probability of getting it with the fair coin
- we want it either way so lets add the 2 which we have already
- done on our calculator so if you add that number to that number
- you get 0.08425 -- its keeps going -- I am just going to round it
- so this is equal to8.425% or 8.43% chance of getting 4 heads in a row
- and once again that's a slightly
- higher number than if all the coins were fair
- because there's a 3/4 chance that I get a coin that has
- a better than even chance of getting heads
- so thats why this number is going to be a little bit higher than the
- probability if I had a fair coin of just getting 4 heads in a row
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