Question

1. if you draw a marble, do not replace it, and draw again from the same bag (4 blue, 5 yellow, 3 red), find the probability of both yellow.

a) $\frac{1}{9}$
b) $\frac{5}{36}$
c) $\frac{1}{11}$
d) $\frac{5}{33}$

1. if you draw one card from a standard 52 - card deck, find the probability of an 8.

a) $\frac{1}{4}$
b) $\frac{1}{13}$
c) $\frac{1}{26}$
d) $\frac{4}{13}$

1. if you draw one card from a standard 52 - card deck, find the probability of a diamond.

a) $\frac{1}{4}$
b) $\frac{4}{13}$
c) $\frac{1}{26}$
d) $\frac{1}{13}$

1. if you draw one card from a standard 52 - card deck, find the probability of a 10 or a spade.

a) $\frac{4}{13}$
b) $\frac{3}{13}$
c) $\frac{1}{4}$
d) $\frac{5}{13}$

Explanation:

Step1: Calculate total number of marbles

Total marbles = \(4 + 5+3=12\)

Step2: Probability of first yellow marble

Probability of first yellow marble \(P(Y_1)=\frac{5}{12}\)

Step3: Probability of second yellow marble

After drawing one yellow marble, number of yellow marbles is \(4\) and total marbles is \(11\). So probability of second yellow marble \(P(Y_2)=\frac{4}{11}\)

Step4: Probability of both yellow (using multiplication rule for dependent events \(P(A\cap B)=P(A)\times P(B|A)\))

\(P = \frac{5}{12}\times\frac{4}{11}=\frac{5\times4}{12\times11}=\frac{20}{132}=\frac{5}{33}\)

Step1: Number of 8 - cards in a deck

There are 4 cards of 8 (one for each suit) in a 52 - card deck.

Step2: Probability formula \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\)

\(P=\frac{4}{52}=\frac{1}{13}\)

Step1: Number of diamond - cards in a deck

There are 13 diamond cards in a 52 - card deck.

Step2: Probability formula \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\)

\(P=\frac{13}{52}=\frac{1}{4}\)

Step1: Use the formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)

Let \(A\) be the event of drawing a 10 and \(B\) be the event of drawing a spade.
Number of 10 - cards \(n(A) = 4\), number of spade - cards \(n(B)=13\), number of 10 of spades \(n(A\cap B) = 1\)

Step2: Calculate \(P(A)\), \(P(B)\) and \(P(A\cap B)\)

\(P(A)=\frac{4}{52}\), \(P(B)=\frac{13}{52}\), \(P(A\cap B)=\frac{1}{52}\)

Step3: Calculate \(P(A\cup B)\)

\(P(A\cup B)=\frac{4 + 13-1}{52}=\frac{16}{52}=\frac{4}{13}\)

Answer:

d) \(\frac{5}{33}\)