Probability question..........?

Question

A tetrahedral has four faces numbered 1,2,3, and 4. Whenthe die is thrown the score is the number on the bottom face. The die is biased so that the probability of scoring a 1 is x, of scoring a 2 is 2x, of scoring a 3 is 3x, and of scoring a 4 is 4x.

(a) Show that x=1/10

The die is to be thrown twice.

(b) Show that the probability of a total score of 7 is 0.24.

(c) Find the probability of scoring a total of 2, 3, 4, 8 WHEN THE DIE IS THROWN TWICE.

(d) Calculate the probability that in two throws of the die the total score will be less than 5.

Answer 1

a. x + 2x + 3x + 4x = 1

    10x = 1

    x = 1/10

b. 0.3 x 0.4 + 0.4 x 0.3 = 0.24

   (to get 7 u either get 3 on the first throw, then 4 on the second, or vice versa)

c. 1-P(total=5,6or7) = 1 - 0.24 - (0.2x0.4+0.4+0.2+0.3x0.3) - (0.2x0.3+0.3x0.2+0.1x0.4+0.4x0.1) = 0.31

d. P(total = 2,3 or 4) = P(total=2,3,4or8)-P(total=8) = 0.31-0.4x0.4=0.15

Answer 2

the sum of all the probabilities should equal to 1

a) let X- " scores on the bottom of the die"

hence, x can equal to 1, 2, 3 or 4

x + 2x +3x + 4x = 1

10x = 1

x = 1/10 ---- SHOWN!!!

b) no of ways of obtaining a seven = (4, 3) or (3, 4)

P(total score is 7) = (4/10 * 3/10) + (3/10 * 4/10) = 0.24 --- SHOWN!!!!

c) No of ways of getting a 2 = (1,1)

No of ways of getting a 3 = (2, 1) OR (1,2 )

No of ways of getting a 4 = (2,2) or (3,1) or (1,3)

no of ways of getting an 8 = (4,4)

P(total score is 2,3,4,8 ) = (1/10 *1/10)+ (2/10*1/10 + 1/10*2/10) +(2/10*2/10 + 3/10 *1/10 + 1/10*3/10) + (4/10*4/10)

= 0.31

d) P(total score less than5) = P(score of 2, 3, 4)

= (1/10*1/10) + (2/10 * 1/10 + 1/10 *3/10 ) + (2/10 * 2/10 + 3/10 * 1/10 + 1/10*3/10)

= 3/20

=0.15

Answer 3

qa

1 = x + 2x + 3x + 4x = 10x

x = 1 / 10

qb

P(S=7)

= P(3)*P(4) + P(4)*P(3)

= 2 * (3/10) * (4/10)

= 0.24

qc

P(S=2) = P(1)*P(1) = (1/10)^2 = 0.01

P(S=3) = 2*P(1)*P(2) = 2 * (1/10) * (2/10) = 0.04

P(S=4) = P(2)*P(2) + 2*P(1)*P(3) = (2/10)^2 + 2 * (1/10) * (3/10) = 0.1

P(S=8) = P(4)*P(4) = (4/10)^2 = 0.16

P(S=2or3or4or8) = 0.01 + 0.04 + 0.1 + 0.16 = 0.31

qd

P(S<5)

= P(S=2or3or4)

= 0.01 + 0.04 + 0.1

= 0.15

Answer 4

am i do ur homework for you, anyways ill try to explain so that i don't feel guilty about giving you the easy way out

a)  when throwing a dice the probability of having either 1 , 2 , 3 or 4 is

   (basically the probability of the dice landing on any number) is the total of weights and since the dice has to land on a number (any number) we know that the total equals to 1

so 1 = 1x + 2x + 3x + 4x

     1 = 10x

so  x = 1/10

b) probability of a total of 7 in two throws is

  

first of all there are 2 scenarios where this can happen

first throw: 3            second throw:4

first throw:4             second throw:3

the probabiliy of the first scenario is 3x X 4x

and the probability of the second scenario is 4x X 3x

and since we dont care which scenario works as long as the total is 7

we can add the probabilities of both scenarios

so          (3x X 4x) + (4x X 3x)  and since x = 1/10

              (3/10 X 4/10) + (4/10 X 3/10) = 12/100 + 12/100 = 24/100=0.24

c) for a total of 2 : basically same method as b but keeping in mind that the only way we can get a total of 2 from two throws is that if we get a 1 on the first and a 1 on the second so 1/10 X 1/10= 0.01

for a total of 3

two scenarios :    1 then 2 or 2 then 1 so (1/10X2/10)+(2/10X1/10)

=0.04

when 4 is the total needed

3 scenarios

1 then 3

3 then 1

2 then 2

so (1/10 X 3/10) + (3/10 X 1/10) + (2/10 X 2/10) = 0.03 + 0.03 + 0.04

= 0.1

when the total needed is 8

only one scenario 4 then 4

so 4/10 X 4/10 = 16/100 = 0.16 ( probability is higher than the previous case although this only works in one certain scenario because the weight of 4 is very high , 4x)

d) probability of getting a total less than 5 is

probability of a total of  1(which is 0 because u cannot have a total of 1 from 2 throws) + probability of a total of 2 + probability of a total of 3 + probability of a total of 4

the probabilities of 2,3,4 are calculated in the previous sections

so probability of a total of less than 5 is =  0.01 + 0.04 + 0.1= 0.15

i hope you got the reasoning behind it too

good luck in math