The probability of rolling a 6 on a biased dice is 1/5. A) complete the tree diagram. B) work out the probability of rolling exactly one six.

a) See attachment

b) P ( exactly one six ) = 0.0256

Step-by-step explanation:

Solution:-

- To develop a probability tree diagram for rolling a biased die 2 times.

- The probability of getting a 6 on each roll is constant and independent between the trials ( rolls ).

- We will consider 2 general probability:

             Probability of getting a 6, p = 1/5

             Probability of not getting a 6, q = 4/5

- For the first roll there are two branches. ( p and q )

- The second roll will entail sub branches for each outcome in first roll.

So, after rolling a 6. We again have two possibilities of either rolling another 6 or not getting a 6.

                            OR

So, after rolling any number other than 6. We again have two possibilities of either rolling a 6 or not getting a 6.

- So, the second roll will have a total of 4 possibilities with the probability of each outcome remains constant and independent from each roll.

        The tree diagram is given as an attachment.

B)

- Once we have developed a tree diagram. We will use it to determine the probability of getting exactly one six out of two rolls.

- First we determine the "paths" that have only one 6. We have 2 paths as follows:

              Path 1: ( First throw = 6 , Second throw = Not getting a 6 )

              Path 2: ( First throw = Not getting a 6 , Second throw =  6 )

- Now we will determine the probabilities for each path.

Note: Remember that while using a tree diagram we multiply all the probabilities along the same path and sum the resultant probabilities of different paths.

Path 1:

                P ( Getting a 6 )*P ( Not getting a 6 )

                p*q

                ( 1 / 5 )*( 4 / 5 )

              = ( 4 / 25 )

Path 2:

                P ( Not getting a 6 )*P ( 6 )

                q*p

                ( 4 / 5 )*( 1 / 5 )

              = ( 4 / 25 )

- Now add the probabilities of different paths to determine the probability of getting exactly one 6.

P ( Exactly one 6 ) = Path 1 + Path 2

                              = ( 4 / 25 ) + ( 4 / 25 )

                              = ( 16 / 625 )

                              = 0.0256     ( answer )

the correct answer is the second option