Question

A road crosses a railroad track at the top of a steep hill.The train cannot stop for oncoming cars, and cars cannot see the train until it is too late. Suppose a train begins crossing the road at time t1 and that the car begins crossing the track at time t2, where 0 < t1 < T and 0 < t2 < T

Required:
a. Find the sample space of this experiment.
b. Suppose that it takes the train al seconds to cross the road and it takes the car a2 seconds to cross the track. Find the set that corresponds to a collision taking place.
c. Find the set that corresponds to a collision is missed by 1 second or less.

Answer 1

a. Sample space = (t₁ , t₂ )

b.A₁ = { t₂ < t₁ < t₂ + a₂ }

A₂ = { t₁ < t₂ < t₁ + a₁ }

if t₁ - t₂ < a₂ , t₂ - t₁ < a₁

c. A₃ = { t₂ + a₂ < t₁ < t₂ + a₂ + 1 }

A₄ = { t₁ + a₁ < t₂ < t₁ + a₁ + 1 }

Explanation:

As given,

A train begins crossing the road at time t₁

The car begins crossing the track at time t₂

where 0 < t₁ < T and 0 < t₂ < T

a.)

Sample space = 0 < t₁ < T , 0 < t₂ < T

b.)

Given that -

The train takes a₁ seconds to cross the road

The car takes a₂ seconds to cross the track.

So, The set that correspond to collision is

A₁ = { t₂ < t₁ < t₂ + a₂ }

A₂ = { t₁ < t₂ < t₁ + a₁ }

if t₁ - t₂ < a₂ , t₂ - t₁ < a₁

c)

The set that corresponds to a collision is missed by 1 second or less is -

A₃ = { t₂ + a₂ < t₁ < t₂ + a₂ + 1 }

A₄ = { t₁ + a₁ < t₂ < t₁ + a₁ + 1 }