There is a large cluster b centered around the origin (0,0), with 8000 points uniformly distributed in a circle of radius 2. there are two small clusters, a and c, each with 1000 points uniformly distributed in a circle of radius 1. the center of a is at (-10,0) and the center of c is at (10,0). suppose we choose three initial centroids x, y, and z, and cluster the points according to which of x, y, or z they are closest. the result will be three apparent clusters, which may or may not coincide with the true clusters a, b, and c. say that one of the true clusters is correct if there is an apparent cluster that consists of all and only the points in that true cluster. assuming initial centroids x, y, and z are chosen independently and at random, what is the probability that a is correct? what is the probability that c is correct? what is the probability that both are correct? after computing these probabilities, identify the true statement from the list below. note: probabilities are rounded to the nearest percent.

The probability of getting zero heads in six tosses is 0.015625.

The probability of getting exactly two heads in six tosses is 0.234375.

Step-by-step explanation:

Using the Minitab software for computing the desired probabilities.

We take n=6 and p=1/2  because we have six tosses and in binomial distribution the probability of success p remains constant in each trial whereas the probability of success in this case is getting heads.

When a coin is tossed then there are two possible outcomes head or tail.

So,

p= P(heads)=1/2=0.5  

The probability of getting zero heads in six tosses is computed by considering the following steps:

In Minitab

Calc >Probability Distributions > Binomial

Select n=6 and p=0.5 and select input constant=0 and bubble the probability and by clicking OK we get the following output:

Probability Density Function  

Binomial with n = 6 and p = 0.5

x  P( X = x )

0    0.015625

So, the probability of getting zero heads in six tosses is 0.015625.

The probability of getting exactly two heads in six tosses is computed by considering the following steps :

In Minitab

Calc >Probability Distributions > Binomial

Select n=6 and p=0.5 and select input constant=2 and bubble the probability and by clicking OK we get the following output:

Probability Density Function  

Binomial with n = 6 and p = 0.5

x  P( X = x )

2    0.234375

So, the probability of getting exactly two heads in six tosses is 0.234375.