independence and expected value - lehi math -...
TRANSCRIPT
Independence and
Expected ValueLESSON 11.4
Objective
• Determine if two events are independent
• Find expected values
Independent vs. Dependent
Independent – an event occurring has no change to future probabilities
Dependent – an event occurring changes future
probability
Independent vs. Dependent
1.
a. You flip a coin and get ‘heads’. Are you more
likely to get ‘heads’ or ‘tails’ on second flip?
b. Which is more likely when flipping a coin 4 times:
HHHH HTHT
Independent vs. Dependent
Independence can be determined if
𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴 ⋅ 𝑃(𝐵)In other words, if the probability of the intersection
(both happening) is the same as the probability of
them happening one after the other, they are
independent.
Independent vs. Dependent
2. A coin is tossed 3 times.
X: the first toss is heads
Y: the second toss is heads
Z: there are exactly 2 consecutive heads
Are the following events independent?
a. X ∩ Y b. X ∩ Z c. Y ∩ Z
Independence:
𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴 ⋅ 𝑃(𝐵)
Independent vs. Dependent
3.
Determine if the events that a customer orders
Landen’s club sandwich and the customer orders
country white bread are independent.
Independent vs. Dependent
4. Town Cinema is next to Rhiannon’s Bistro. 200
people were surveyed who saw a movie and ate at
the bistro. 80% liked the movie while 60% liked both
the movie and the meal. Assuming independence,
what is that probability that a randomly chosen
moviegoer will like a meal at the bistro?
Conditional Independence
CONDITIONAL PROBABILITY
If 𝑃 𝐵 𝐴 = 𝑃(𝐵), then A did not change the
probability of B, making A and B independent.
Independent vs. Dependent
5. Hamid rolls a 6-sided die 3 times.
A: the first roll is an odd number
B: there are exactly 2 consecutive odd numbers
Are A and B independent events? Show using
conditional probability.
Independent vs. Dependent
6. TEN: student is in 10th grade
TWELVE: student is in 12th grade
FT: student prefers field trip
TS: student prefers talent show
Find P(TEN|FT) and P(TEN). Now find P(FT|TEN) and P(FT).
Independent vs. Dependent
7. M: student selects milk
B: student selects burger
Are events M and B independent?
Expected Value
Using probability, we can find an EXPECTED VALUE.
EX: 200 people sign up for an event but only 80% are
expected to show up. The event staff should plan
for 200 × 80% → 200 × .80 = 160 people to show up.
Expected Value
8. A two-year warranty on a bike covers the bike frame and
the gear shifter, fixing either for free. Without a warranty, fixing
a frame is $225 and a gear shifter is $110.
Madelyne buys a bike. Over a two year period,
P(frame breaking)=0.15 and P(gear shifter breaking)=0.30.
The warranty is $75. Should Madelyne buy the warranty?
Homework
Worksheet 11.4