Consider the following random experiment. First, roll a die and observe the number of dots facing up; then, toss a coin the number of times that the die shows and observe the total number of heads. For example, if the die shows three dots facing up and the coin (which is then tossed three times) comes up heads exactly twice, then the outcome of the experiment can be represented as (3, 2). a. Determine a sample space for this experiment.
b. Determine the event that the total number of heads is even.

{(1, 0), (1, 1), (2,0), (2,1), (2, 2), (3,0), (3,1), (3,2), (3,3), (4, 0), (4,1), (4,2), (4,3), (4,4), (5,0), (5,1),(5,2), (5,3), (5,4), (5,5), (6,0), (6,1),(6,2), (6,3), (6,4), (6,5), (6,6)}

B.)

{(2,2), (3,2), (4,2), (4,4), (5,2), (5,4), (6,2), (6,4), (6,6)}

Step-by-step explanation:

Sample space :

Die roll, then number of coin tosses based on the outcome of the die roll, then number of heada from the toss(es) is recorded along with the outcome of the die roll.

{(1, 0), (1, 1), (2,0), (2,1), (2, 2), (3,0), (3,1), (3,2), (3,3), (4, 0), (4,1), (4,2), (4,3), (4,4), (5,0), (5,1),(5,2), (5,3), (5,4), (5,5), (6,0), (6,1),(6,2), (6,3), (6,4), (6,5), (6,6)}

Event that the total number of head is even:

{(2,2), (3,2), (4,2), (4,4), (5,2), (5,4), (6,2), (6,4), (6,6)}

step-by-step explanation:

try remembering the steps

∠1≅∠2 by the alternate exterior angles theorem.

step-by-step explanation:

given, a ∥ b and ∠1 ≅ ∠3 .we have to prove that e ∥ f

we know that ∠1≅∠3 and that a || b because they are given. we see that by the alternate exterior angles theorem. therefore, ∠2≅∠3 by the transitive property. so, we can conclude that e || f by the converse alternate exterior angles theorem.

we have to fill the missing statement.

transitivity property states that if a = b and b = c, then a = c.

now, given ∠1≅∠3 and by transitivity property ∠2≅∠3 .

hence, to apply transitivity property one angle must be common which is not in result after applying this property which is ∠1.

the only options in which ∠1 is present are ∠1 and ∠2, ∠1 and ∠4

∠1 and ∠4 is not possible ∵ after applying transitivity we didn't get ∠4.

hence, the missing statement is ∠1≅∠2.

so, ∠1≅∠2 by the alternate exterior angles theorem.