A probability experiment: spin a spinner with equal sections 1,2,3,4,5 and draw a card from a standard deck and note its suit (hearts, diamonds, clubs, spades). List the sample space (use tree mentally). How many outcomes?
Counting: 5 spinner outcomes ร 4 suits = 20 outcomes.Sample space = {1โฅ,1โฆ,1โฃ,1โ , 2โฅ,...,5โ }. (explicit list not needed)
3.1.2 โ Simple event? picking a Jack or a heart?
From a standard deck, consider event C = โthe card is a Jack or a heartโ. Is C a simple event? Explain.
No โ it contains multiple outcomes: all 4 Jacks plus the hearts that are not Jack (12 hearts) โ total 4+12=16 outcomes. Simple events have exactly one outcome.
3.1.3 โ Pizza combinations
A pizza shop offers: crust (thin, thick, stuffed) ; sauce (red, pesto, BBQ) ; cheese (mozzarella, vegan, blend) ; topping (pepperoni, mushrooms, onions, sausage). How many different pizzas can you make with exactly one choice from each category?
3.1.4 โ Two fair dice, find P(sum = 7) and P(sum > 9)
Roll two sixโsided dice (one red, one blue). Use classical probability.
Total outcomes = 36. Sum=7: {(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)} โ 6 โ 6/36=1/6โ0.1667. Sum>9: sum 10,11,12. (4,6),(5,5),(6,4),(5,6),(6,5),(6,6) โ 6 outcomes โ 6/36=1/6.
3.1.5 โ Laptop brand preference (empirical)
Brand
frequency
Dell
84
HP
63
Apple
97
Lenovo
42
Other
14
If one respondent is chosen at random, what is the empirical probability that they prefer Apple or Dell?
Total = 84+63+97+42+14 = 300. Apple+Dell freq = 97+84=181 โ P = 181/300 โ 0.6033.
3.1.6 โ Classify probability type
a) โThe chance that it will rain tomorrow is 30%โ โ meteorologistโs forecast.
b) โThe probability of drawing a red card from a deck is 1/2.โ
c) โBased on records, the probability a patient tests positive for a certain disease is 0.02.โ
a) subjective (educated forecast, not longโrun frequency). b) classical (equally likely). c) empirical (from historical data).
3.1.7 โ Complement rule (flight delays)
At a certain airport, the probability that a flight is delayed is 0.18. Find the probability that a flight is not delayed.
A bag contains 3 red marbles and 5 blue marbles. You draw one marble (without replacement, but it doesn't affect coin), then toss a fair coin. Use a tree to find P(red and tails).
P(red) = 3/8. P(tails) = 1/2. independent? marble draw does not affect coin โ multiply: (3/8)*(1/2) = 3/16 = 0.1875.
3.1.9 โ License plate probability
A state license plate has 3 letters followed by 3 digits (letters AโZ, digits 0โ9). Letters and digits can repeat. What is the probability that a randomly generated plate reads โABC 123โ?
Total plates = 26ยณ ร 10ยณ = 17,576 ร 1,000 = 17,576,000. Only one specific plate โ P = 1 / 17,576,000 โ 5.69e-8.
๐ 3.2 โ fresh conditional & multiplication
3.2.1 โ Candies: two draws without replacement
A jar contains 6 strawberry candies and 4 orange candies. You draw two candies without replacement. Find P(second is orange | first is strawberry).
After removing one strawberry: left 5 strawberry + 4 orange = 9 total. P(orange|first strawberry) = 4/9 โ 0.444.
3.2.2 โ Smartphone vs age group
iPhone
Android
Other
under 30
45
32
5
30 or older
28
54
16
Find P(iPhone | under 30).
Total under 30 = 45+32+5 = 82. iPhone under30 = 45 โ 45/82 โ 0.549.
3.2.3 โ Independent? marble with replacement
Draw a marble from a bag (3 red, 5 blue), note color, replace it, then draw again. Are the events โfirst redโ and โsecond blueโ independent?
Yes, because replacement restores original composition. P(second blue|first red) = P(blue) = 5/8 โ independent.
3.2.4 โ P(king then queen) with replacement
Two cards drawn from a deck with replacement. Find P(king first and queen second).
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