A fair die is rolled 10 times. What is the probability that an odd number (1, 3, or 5) will occur less than 3 times?

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Answer 1 · 2017-10-21T19:29:28

# 7/128.#

Let us call, getting an odd no. on a roll of die, Success.

The, clearly, the probability #p# of Success is #3/6=1/2.#

Hence, #q=1-p=1/2.#

If, #X=x# denotes the no. of success in #n# trials, then, #X# is a

Binomial Random Variable, with parameters

#n=10, and, p=1/2.#

Then, the Probabilty of #x# success out of #n# trials, i.e., #P(X=x),# is,

#P(X=x)=p(x)=""_nC_xp^x,q^(n-x), x=0,1,2,...,n.# In our case,

#P(X=x)=""_10C_x(1/2)^x(1/2)^(10-x), x=0,1,2,...10, i.e.,#

#P(X=x)=p(x)=""_10C_x(1/2)^10=(""_10C_x)/1024, x=0,1,...,10.#:.Hence,

#:. "The Reqd. Prob.="P(X<3),#

#=P(X=0)+P(X=1)+P(X=2),#

#=1/1024{""_10C_0+""_10C_1+""_10C_2},#

#=1/1024{1+10+45},#

#=56/1024.#

#rArr "The Reqd. Prob.="7/128.#

Enjoy Maths.!