Answers should be exact decimals (please do not leave as fractions or unfinished

Question

Answers should be exact decimals (please do not leave as fractions or unfinished calculations). A six-sided die (cube) with faces numbering 1,2,3,4,5,6 is loaded so that the probability of a six is 1/2. All the remaining numbers are equally likely. 1) Find the probability that... ...rolling the die results in a 3 or a 6: ...rolling the die results in not getting a 6: ..rolling the die results in the number being either greater than 2 or less than 5: ..rolling the die results in an even number or a 5: .rolling the die twice results in a 6 followed by a 4: ...rolling the die three times results in all 6's: ...it takes exactly six rolls to get a number that's NOT a 6: 2) We roll the loaded die four times. Find the probability that... ...all the rolls are 2: ...none of the rolls are 2: ...exactly one of the rolls is a 2: ...at least one of the rolls is a 2: ..either the first roll or the last roll is a 2:

Explanation / Answer

Solution:

Probability of Gettting a 6 = 1/2

So probability of not getting 6 = 1-1/2 = 1/2

Probability of getting 1 = (1/2)*(1/5) = 1/10

Probability of getting 2 = (1/2)*(1/5) = 1/10

Probability of getting 3 = (1/2)*(1/5) = 1/10

Probability of getting 4 = (1/2)*(1/5) = 1/10

Probability of getting 5 = (1/2)*(1/5) = 1/10

Probability of getting 6 = (1/2)*(1/5) = 1/2

Solution:

Probability of Rolling a die getting 6 or 3 = (1/2)+(1/10) = 6/10 = 3/5 or 0.6

Probability of resulting not getting 6 = 1-1/2 = 1/2 or 0.5

Probability of resulting a getting greater than 2 and less than 5 = P(X=3) + P(X=4) = (1/10) + (1/10) = 2/10 = 1/5 = 0.2

Probability of even no or 5 = P(X=2)+P(X=4)+P(X=5)+P(X=6) = 1/10+ 1/10 + 1/10 +1/2 = 8/10 = 4/5 = 0.8

Probability of getting 6 followed by 4 = P(X=6) * P(X=4) = (1/2)*(1/10) = 1/20 = 0.05

Probability of getting 6 in rolling three times die = (1/2)*(1/2)*(1/2) = 1/8

Probabilty of getting a no. i.e. not a 6 = P(X=6) *P(X=6) *P(X=6) *P(X=6) *P(Xnot equal 6) = (1/2)*(1/2)*(1/2)*(1/2)*(1/2)*(1/2) = 1/64

Solution2:

All the rolls are 2 = (1/10)*(1/10)*(1/10)*(1/10) = 1/10000 = 0.0001 or 0.01%

None of the rolls are 2 = (9/10)*(9/10)*(9/10)*(9/10) = 0.6561 or 65.61%

Exactly one of the rolls is a 2 = ((1/10)*(9/10)*(9/10)*(9/10)) +((9/10)*(1/10)*(9/10)*(9/10))+((9/10)*(9/10)*(1/10)*(9/10))+((9/10)*(9/10)*(9/10)*(1/10)) = 0.0729 * 4 = 0.2916 or 29.16%

At least one of the roll is a 2 = (1/10)*(9/10) *(9/10)*(9/10) + (1/10)*(1/10)*(9/10)*(9/10) + (1/10)*(1/10)*(1/10)*(9/10) +(1/10)*(1/10)*(1/10)*(1/10) = 0.0729 + 0.0081 + 0.0009 + 0.0001 = 0.082 or 8.2%

Either the first roll or the last roll is a 2 = (1/10)*(9/10)*(9/10)*(9/10) + (9/10) *(9/10) *(9/10) * (1/10) = 2 * 0.0729 = 0.1458 or 14.58%

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