Question

Please provide solutions to the following problems. Please use Excel to solve the problems and submit the Excel spreadsheet.

A fair coin is tossed 15 times, calculate the probability of getting 0 heads or 15 heads A biased coin with probability of head being .6 is tossed 12 times. What is the probability that number of head would more than 4 but less than or equal to 10. You have a biased dice (with six faces numbered 1,2,3,4,5 and 6) in that odd numbers are thrice as likely as even numbers to show. You toss the dice 10 times. What is the probability that an even number would show up 5 times. If z has a standard normal distribution, calculate the following probability. P(-.3≤z≤10) If x has a normal distribution with mean 15 and standard deviation 2, calculate the following probabilities: P(1≤x<15)

Answer 1

Let A: the event that head appears. and suppose a head is considered as a 'success' .

Let X be the number of heads obtained. Then X~ Bin(n,p)

1. FAIR COIN :

p= P[A] = 0.5 , n= 15

So, P[X=0]=

= 3.05 10^-5

P[X=15] =

= 3.05 10^-5

BIASED COIN :

p= 0.6 , n = 12

So, P[4<X<=10]= P[X=5]+P[X=6]+.....+P[X=10]

= ++..........+

BIASED DICE:

Let A: the event that an even number appears

and B: the event that an odd number appears.

Let, P[A]= p , then ac to question, P[B]= 3p;

We know. P[A]+ P[B]= 1

or, p+ 3p = 1

or, p = 1/4 ;

Now, let X_i= 1, if an even number appears in i^th draw, with probability p=1/4

= 0 , if an odd number appears in i^th draw, with probability q= 1-p = 3/4 ; i= 1(1)10 ;

So, Total no of even number ( Y=   ) obtained in 10 draws should follow a binomial distribution with parameters n= 10 and p =1/4 .

So, P[Y=5] = = 0.0584 ,which is the probability of obtaining exact 5 even numbers.

Though if you want at least 5 even numbers, the probability would be

P[Y>=5] = 1 - P[Y<5]

=1- P[Y=4]-P[Y=3]-P[Y<2] -P[Y=1]-P[Y=0]

STANDARD NORMAL:

Z~ N(0,1)

then P[-0.3<Z<10]= P[Z<10]- P[Z<-0.3]

= P[Z<10] -1+ P[Z<0.3]

  

= 1-1+0.6179

=0.6179

NORMAL X:

X~N(15,2)

then  

So,  

  

  

= 0.5 - 0

=0.5 (ans)

I dont think this could be done in excel. I hope it will help you.If you have any doubts, let me know in the comment section. Thank you.