Question

Someone gets more than 8 hours of sleep 50 percent of the time, less than 3 hours 30 percent of the time, and 3-8 hours (inclusive) 20 percent of the time. when they sleep more than 8 hours, forty percent of the time they score really well. when they get less than 3 hours, 20 percent of the time they score really well. when they get 3-8 hours inclusive they score really well 40 percent of the time. what is their overall probability of scoring really well?

Answer 1

Someone gets more than 8 hours of sleep 50 percent of the time.

less than 3 hours of sleep 30 percent of the time.

3-8 hours of sleep 20 percent of the time.

When sleep is more than 8 hours, 40% of the time they score really well.

When sleep is less than 3 hours, 20% of the time they score really well.

When sleep is 3-8 hours, 40% of the times they score really well.

So, the given facts are

P(Sleep more than 8 hours)=0.5.

P(Sleep less than 3 hours)=0.3.

P(Sleep 3-8 hours)=0.2.

P(Scoring well|Sleep more than 8 hours)=0.4.

P(Sleep less than 3 hours)=0.2.

P(Scoring Well|Sleep 3-8 hours)=0.4.

So, by total probability theorem, the probability of scoring well is

=P(Scoring well)

=P(Sleep more than 8 hours)*P(Scoring well|Sleep more than 8 hours)+P(Sleep less than 3 hours)*P(Sleep less than 3 hours)+P(Sleep 3-8 hours)*P(Scoring Well|Sleep 3-8 hours)

=0.5*0.4+0.3*0.2+0.2*0.4

=0.2+0.06+0.08

=0.34

So, the probability of scoring weell, is 0.34.