In a fair coin, the chances are 50%:
A loaded coin the one that comes up with one of the two much more frequently than the other.
What is the probability of if 2 flips are heads?
Suppose we flip our coin twice. What we care about is that exactly one of the two things is heads, and thereby exactly the other one is tails. For a fair coin, what do you think the probability would be that if I flip it twice we would see heads exactly once?
Given that, we now have to associate a truth table with the question we're asking. So where exactly is, in the outcome, heads represented once?
What is the probability that exactly 1 of those 3 flips comes up heads?
Answer
Here is a challenging question. I'll give you a loaded coin--the probability for H is 0.6. I expect this will take you awhile on a piece of paper to really calculate this probability over here. But you can do exactly the same thing. You go through the truth table. You apply the multiplication I showed you before to calculate the probability of each outcome; they're not the same anymore. H, H, H is clearly more likely than T, T, T. And when you've done this, add the corresponding figures up, and tell me what the answer is.
In [17]:
from itertools import product
from numpy import prod
heads = 0.6
tails = 0.4
truth_table = list(product((heads, tails), repeat=3))
for x in truth_table:
print x, prod(x)
In [20]:
result = []
for x in truth_table:
if x.count(0.6) == 1:
print x, prod(x)
result.append(prod(x))
print sum(result)
Now I am throwing dice. The difference between dice and coins is that there are now 6 possible outcomes. Let me just draw them, and say it's a fair die, which means each of the different sides comes up with a probability over 6 for any of the numbers you can plug in over here. What do you think the probability is the die comes up with an even number? I'm going to write this as the outcome of the die is even. And you can once again use a truth table to calculate that number.
Answer
Suppose we throw a fair die twice. What do you think the probability of a double is? Double means both outcomes are identical with the same number regardless of what that number is. The actually an important number because in many games involving two dice, have different rules when these come up with the same number. So, it might be important to know what the probability is.
Answer
In our next unit, we will talk about dependence, where there are bizarre dependencies between different outcomes. But for the time being, you really managed to get a very basic understanding of probability.