Python Tutorial : Probability mass and distribution functions

Key Takeaways

after conducting many random experiments you will notice that some outcomes are more likely than others this is called a probability distribution there are two important functions that are useful for probability calculations the probability mass function and the cumulative distribution function a discrete random variable has a finite number of possible outcomes the probability mass function allows you to calculate the probability of getting a particular outcome for a discrete random variable the binomial probability mass function allows you to calculate the probability of getting K heads from n coin flips with the probability of getting helps the formula multiplies the number of different ways you can get K successes out of n coin flips by the probability of success raised to the number of successes k by the probability of failure 1 minus P raised to the number of failures and minus K it's okay if you don't understand the formula right now with practice your intuition about this will grow if we plot the probability mass function of getting K heads out of 10 third coin flips you can see that 5 is the most likely outcome with the SyFy dot stats library we can use the Bynum the PMF function to calculate this probability if you use Bynum PMF with parameters K equals 2 N equals 10 and P equals 0.5 to get the probability of getting two heads from 10 flips of a fair coin that is 4% the probability of getting five heads from ten coin flips is almost 25% the probability of getting 50 heads out of 100 flips of a biased coin with 30% probability of getting heads is extremely small not even a 1% chance if instead you calculate the probability of getting sixty five heads from 100 flips of a biased coin with 70 percent probability of getting heads you see that it's almost five percent as n gets larger the probability of getting K heads become smaller for the same P if you instead want to calculate the probability of getting K or fewer heads from n throws you use the binomial probability distribution function which adds the probability of getting zero heads out of n flips getting heads once out of n flips and getting all the way up to K heads out of n flips the binomial probability distribution function allows us to calculate the cumulative probability of getting K heads or fewer from n coin flips with P probability of getting heads in Python we use the venom the CDF function with parameters K N and P adding the probabilities from the mass function we get the cumulative distribution function CDF this is a way of getting a range of probabilities rather than the probability of a single event with the SyFy the stats library we can use the binomial function to get such a probability using the same parameters if you use the binomial CDF with parameters K equals 5 N equals 10 and P equals 0.5 you get the probability of getting heads 5 times or fewer out of 10 flips which is 62% the probability of getting heads 50 times or fewer out of 100 flips a biased coin with 30% probability of getting heads is near 100% it's almost guaranteed the probability of getting heads more than fifty nine times from 100 flips of a biased coin with P equals 70% is 99% again it's almost certain what if we want the probability of getting heads more than K times this is called the complement and we get it by subtracting the CDF from one alternatively we can calculate the complement using the function Bynum dot s F with the same parameters s f stands for survival function which allows you to get tail probabilities or the complement in this case we've had some fun calculating probabilities now let's pray