Example Coin-flipping

heads (0) & tails (1). For 100 trials, what is the probability that there will be less than 4 heads?

p=0.5; N=100;

$$[1 + N + N(N-1)/2 + N(N-1)(N-2)/3]0.5^N= 328451 \times 0.5^{100} = 2.6 \times 10^{-25}$$

The chance of getting less than 4 heads is practically zero: if the remaining life time of our solar system is 10 billion years (t= $3.15 \times 10^{17}$s) and it takes 1s to sample each set of 100 throws, the chance of obtaining 4 or less heads is $P_r(X<=4) =8 \times 10^{-8}$.

  

Figure 3.3: An example of a Binomial distribution curve. [stats_uib_3_3.m]

Fig. 3.3 shows the theoretical (Binomial) probability distribution function for 100-throw experiments.