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Each trial (for example, each coin toss) is completely independent of the results of the previous turn. There can be only two outcomes – success/yes and failure/no. The probability of either outcome remains constant from trial to trial. For example, the probability of landing heads in a coin toss remains 50% regardless of what happened in a previous coin toss.

Mathematically, if we say that the probability of success in a Bernoulli trial is p, then the probability of failure in the same trial, q, can be written as: q=1–p Thus, in a coin toss experiment, if probability of landing heads is 50% or 0.5, then probability of landing tails is: q = 1 – 0.50 q = 0.50 Similarly, let’s consider a dice roll experiment where we consider landing a 6 to be “success” and anything below that to be “failure”. We know that probability is defined as:

Probability of an event =

number of positive outcomes total number of outcomes

Here, number of positive outcomes is 1 and total number of possible outcomes is 6 (since there are six number of a dice). Thus, probability of success p (landing a 6) is 1/6. Based on the above, the probability of failure q can be written as: q = 1 – 1/6 q = 5/6 Using Binomial Probability Formula to Calculate Probability for Bernoulli Trials The binomial probability formula is used to calculate the probability of the success of an event in a Bernoulli trial. Hence, the first thing we need to define is what actually constitutes a success in an experiment. This is completely arbitrary and depends on the experiment itself. One experiment may define it as the chances of rolling a 6 on a dice, another may define it as the chances of rolling 3 or more. Based on this, we have the following formula: Probability of k successes in n trials

n n n−k P( X = k ) =   p k (1 − p ) =   p k q n − k k  k 

Where: n = total number of trials k = total number of successes n – k = total number of failures p = probability of success in one trial q = probability of failure in one trial (i.e. 1 – p)

n n! = binomial coefficient  =  k  k !(n − k )! This model is called "binomial distribution": it uses the binomial coefficient. An example will illustrate this formula better: Example: Calculate the probability of rolling 4 on a dice exactly 5 times in 25 trials. Here, we have the following: n = total trials = 25 k = total successes = 5 n – k = total failures = 20 p = 1/6 = 0.167 q = 5/6 = 0.833

n   = 53130 k  Therefore, probability (P) will be: P (0.00000335872) = 0.17844. Thus, the probability is 0.17844.

5

= 53130 x(0.167) (0.833)

20

= 53130 x (0.0001298)(0.02587) = 53130 x