Question

4 \aardvark\: find each probability if you pick a card, do not replace it, then pick a second card.
o. p(v then k)
t. p(a then r)
e. p(d then r)
y. p(r then not r)
n. p(a then v)
e. p(a then not a)

5 find each probability if you pick two marbles without replacing the first (g = green; r = red; y = yellow)
t. p(red then green)
f. p(yellow then not yellow)
e. p(red then yellow)
r. p(green then not green)
m. p(green then green)
w. p(not red then not red)

Explanation:

Problem 4: "AARDVARK" Card Probabilities (8 total cards: A, A, R, D, V, A, R, K → counts: A=3, R=2, D=1, V=1, K=1)

Step1: Define total & conditional prob

For sequential draws without replacement, $P(\text{Event 1 then Event 2}) = P(\text{Event 1}) \times P(\text{Event 2}|\text{Event 1})$
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O. P(V then R)

Step1: Probability of picking V first

$P(V) = \frac{1}{8}$

Step2: Probability of picking R next

$P(R|V) = \frac{2}{7}$

Step3: Multiply the two probabilities

$P(V \text{ then } R) = \frac{1}{8} \times \frac{2}{7} = \frac{2}{56} = \frac{1}{28}$

T. P(A then R)

Step1: Probability of picking A first

$P(A) = \frac{3}{8}$

Step2: Probability of picking R next

$P(R|A) = \frac{2}{7}$

Step3: Multiply the two probabilities

$P(A \text{ then } R) = \frac{3}{8} \times \frac{2}{7} = \frac{6}{56} = \frac{3}{28}$

E. P(D then R)

Step1: Probability of picking D first

$P(D) = \frac{1}{8}$

Step2: Probability of picking R next

$P(R|D) = \frac{2}{7}$

Step3: Multiply the two probabilities

$P(D \text{ then } R) = \frac{1}{8} \times \frac{2}{7} = \frac{2}{56} = \frac{1}{28}$

Y. P(R then not R)

Step1: Probability of picking R first

$P(R) = \frac{2}{8} = \frac{1}{4}$

Step2: Probability of non-R next

$P(\text{not } R|R) = \frac{6}{7}$

Step3: Multiply the two probabilities

$P(R \text{ then not } R) = \frac{1}{4} \times \frac{6}{7} = \frac{6}{28} = \frac{3}{14}$

N. P(A then V)

Step1: Probability of picking A first

$P(A) = \frac{3}{8}$

Step2: Probability of picking V next

$P(V|A) = \frac{1}{7}$

Step3: Multiply the two probabilities

$P(A \text{ then } V) = \frac{3}{8} \times \frac{1}{7} = \frac{3}{56}$

E. P(A then not A)

Step1: Probability of picking A first

$P(A) = \frac{3}{8}$

Step2: Probability of non-A next

$P(\text{not } A|A) = \frac{5}{7}$

Step3: Multiply the two probabilities

$P(A \text{ then not } A) = \frac{3}{8} \times \frac{5}{7} = \frac{15}{56}$

Step1: Define total & conditional prob

For sequential draws without replacement, $P(\text{Event 1 then Event 2}) = P(\text{Event 1}) \times P(\text{Event 2}|\text{Event 1})$
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T. P(red then green)

Step1: Probability of picking red first

$P(R) = \frac{2}{9}$

Step2: Probability of picking green next

$P(G|R) = \frac{4}{8} = \frac{1}{2}$

Step3: Multiply the two probabilities

$P(R \text{ then } G) = \frac{2}{9} \times \frac{1}{2} = \frac{2}{18} = \frac{1}{9}$

F. P(yellow then not yellow)

Step1: Probability of picking yellow first

$P(Y) = \frac{3}{9} = \frac{1}{3}$

Step2: Probability of non-yellow next

$P(\text{not } Y|Y) = \frac{6}{8} = \frac{3}{4}$

Step3: Multiply the two probabilities

$P(Y \text{ then not } Y) = \frac{1}{3} \times \frac{3}{4} = \frac{3}{12} = \frac{1}{4}$

E. P(red then yellow)

Step1: Probability of picking red first

$P(R) = \frac{2}{9}$

Step2: Probability of picking yellow next

$P(Y|R) = \frac{3}{8}$

Step3: Multiply the two probabilities

$P(R \text{ then } Y) = \frac{2}{9} \times \frac{3}{8} = \frac{6}{72} = \frac{1}{12}$

R. P(green then not green)

Step1: Probability of picking green first

$P(G) = \frac{4}{9}$

Step2: Probability of non-green next

$P(\text{not } G|G) = \frac{5}{8}$

Step3: Multiply the two probabilities

$P(G \text{ then not } G) = \frac{4}{9} \times \frac{5}{8} = \frac{20}{72} = \frac{5}{18}$

M. P(green then green)

Step1: Probability of picking green first

$P(G) = \frac{4}{9}$

Step2: Probability of picking green next

$P(G|G) = \frac{3}{8}$

Step3: Multiply the two probabilities

$P(G \text{ then } G) = \frac{4}{9} \times \frac{3}{8} = \frac{12}{72} = \frac{1}{6}$

W. P(not red then not red)

Step1: Probability of non-red first

$P(\text{not } R) = \frac{7}{9}$

Step2: Probability of non-red next

$P(\text{not } R|\text{not } R) = \frac{6}{8} = \frac{3}{4}$

Step3: Multiply the two probabilities

$P(\text{not } R \text{ then not } R) = \frac{7}{9} \times \frac{3}{4} = \frac{21}{36} = \frac{7}{12}$

Answer:

(Problem 4):

• O. $\frac{1}{28}$
• T. $\frac{3}{28}$
• E. $\frac{1}{28}$
• Y. $\frac{3}{14}$
• N. $\frac{3}{56}$
• E. $\frac{15}{56}$

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Problem 5: Marble Probabilities (9 total marbles: G=4, R=2, Y=3)