Question

a) The probability of getting exactly 2 sets of 3( of the same kind) in a 7-card from a regular deck of cards is:
b)If you roll a fair, 6-sided die 5 times, the probability that the sum of all5 rolls is greater than 7 is:
c) If you roll a 6-sided die 3 times, the probability that the product of the three rolls will be even is:
d) if you flip a fair coin 10 times, the probability of getting exactly 3 heads or exactly 3 tails:

Answer 1

a) The probability of getting exactly 2 sets of three of kind, in a 7 card from a regular deck of cards is computed here as:
= Number of ways to select 2 ranks which would come thrice each * Number of ways to select 1 card from remaining 44 cards / Total ways to select 7 cards from 52 cards

Therefore 0.000410 is the required probability here.

b) Probability that the sum of 5 rolls is greater than 7 is computed here as:
= 1 - Probability that the sum of 5 rolls is 5 or 6 or 7

= 1 - P(11111) - P(all are ones and one is 2) - P(all are 1s and two outcomes are 2) - P(all are ones and one outcome is 3)

= 1 - (1 + 5 + 5c2 + 5) / 6⁵

= 1 - (21/6⁶) = 0.999550

Therefore 0.999550 is the required probability here.

c) The probability that the product of three roll is even is computed here as:

= 1- Probability that all the three numbers are odd

= 1 - (3³) /6³

= 1 - 0.5³ = 0.875
Therefore 0.875 is the required probability here.

d) The probability of getting exactly 3 heads or exactly 3 tails in a flip of 10 coins is computed here as:

= P(X = 3) + P(X = 7)

= 0.5¹⁰*(10c3 + 10c7)

= 0.5¹⁰*2*120 = 0.234375

Therefore 0.234375 is the required probability here.