Dice Probability

Question

A. A coin is tossed and then a pair of dice is rolled. Find the probability that the outcome is a Head and a sum of 8

 

 

B. A coin is tossed 9 times, find the probability that the first 4 are heads and the last five are tails.

 

Answer 1

A.) the probability of getting a Head is 1/2 (and 1/2 for a Tail also)

roll a pair of dice

what are the outcomes ? (or the sample space)

11,12,13,14,15,16

21,22,23,24,25,26

31,32,33,34,35,36

41,42,43,44,45,46

51,52,53,54,55,56

61,62,63,64,65,66

there are 36 possible outcomes

which ones sum to 8 ?

26,35,44,53, and 62 for a total of 5 out of 36, or 5/36

the probability of getting a Head and a sum of 8 is (1/2)(5/36)=5/72

 

B.) if you toss a coin once there are 2 outcomes:H or T (2¹=2)

     if you toss a coin twice there are 4 outcomes: HH, HT, TT, TH (2²=4)

     if you toss a coin three times there are 8 outcomes:

     HHH,TTT,HHT,THH,HTH,TTH,HTT,THT (2³=8)

     if you toss a coin 9 times there are 2⁹=512 outcomes

     there is only one way to get HHHTTTTT, so the probability is 1/512

Answer 2

A.  When you want to find the probability of two subsequent events, you find each event's probability and then  

     multiply the two probabilities.

 

So if we look at all the possible outcomes of tossing a coin it is only two: Heads or Tails.

So the probability is 1/2

 

If we look at all of the possible outcomes of rolling a pair of dice we have to consider each roll of a die as a separate possibility, because the outcome of the first die does not depend on the outcome of the second die.

 

The possible outcomes of rolling one die is 6.  Therefore the probability of rolling a pair of dice is 1/6 x 1/6

 

So to find the probability of all three events we just multiply the probabilities of all events.  1/2 x 1/6 x 1/6

 

B. When you toss nine coins you have 9 separate events with 2 outcomes each.  So the probability of one

    combination of outcomes is (1/2)⁹  (1/2 multiplied by itself 9 times)