Question

determine whether the scenario involves independent or dependent events. then find the probability. express the probability as a reduced fraction, a decimal rounded to the thousandths and a percentage rounded to the hundredths.

1. you flip a coin twice. the first flip lands tails - up and the second flip also lands tails - up.
2. a basket contains four apples, three peaches, and three pears. you randomly select and eat three pieces of fruit. the first piece of fruit is an apple and the next two pieces are peaches.
3. a bag contains four red marbles and five blue marbles. another bag contains five green marbles and six yellow marbles. you randomly pick one marble from each bag. one marble is blue and one marble is yellow.
4. there are ten shirts in your closet, four blue and six green. you randomly select one to wear on monday and then a different one on tuesday. you wear blue shirts both days.
5. there are eleven shirts in your closet, four blue, three green, and four red. you randomly select a different shirt each day. you wear a blue shirt on monday, a green shirt on tuesday, and a red shirt on wednesday.
6. there are twelve shirts in your closet, four blue, four green, and four red. you randomly select a different shirt each day. you wear a blue shirt on monday, a green shirt on tuesday, and a red shirt on wednesday.
7. a box of chocolates contains three milk chocolates, four dark chocolates, and three white chocolates. you randomly select and eat three chocolates. the first piece is milk chocolate, the second is dark chocolate, and the third is white chocolate.
8. there are eight nickels and eight dimes in your pocket. you randomly pick a coin out of your pocket and place it on a counter. then you randomly pick another coin. both coins are nickels.

Explanation:

Let's solve problem 9:

Step 1: Identify Event Type

Flipping a coin twice: the result of the first flip (tails) does not affect the second flip. So, these are independent events. The probability of tails on a fair coin is $P(\text{tails}) = \frac{1}{2}$.

Step 2: Calculate Joint Probability

For independent events, $P(A \cap B) = P(A) \times P(B)$. Here, $A$ is first tails, $B$ is second tails. So:
$$P(\text{tails, tails}) = P(\text{tails}) \times P(\text{tails}) = \frac{1}{2} \times \frac{1}{2}$$

Step 3: Simplify and Convert

Simplify the fraction: $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$.
Decimal: $\frac{1}{4} = 0.25$ (already to thousandths as $0.250$).
Percentage: $0.25 \times 100 = 25.00\%$.

Step 1: Identify Event Type

After eating the first fruit, the total number of fruits decreases, so the events (first apple, then two peaches) are dependent.
Total fruits initially: $4 + 3 + 3 = 10$.

Step 2: Probability of First Event (Apple)

$P(\text{apple first}) = \frac{\text{number of apples}}{\text{total fruits}} = \frac{4}{10}$.

Step 3: Probability of Second Event (Peach)

After eating an apple, total fruits left: $9$. Number of peaches: $3$.
$P(\text{peach second}) = \frac{3}{9}$.

Step 4: Probability of Third Event (Peach)

After eating an apple and a peach, total fruits left: $8$. Number of peaches left: $2$.
$P(\text{peach third}) = \frac{2}{8}$.

Step 5: Multiply for Joint Probability

For dependent events, $P(A \cap B \cap C) = P(A) \times P(B|A) \times P(C|A \cap B)$.
So:
$$P = \frac{4}{10} \times \frac{3}{9} \times \frac{2}{8}$$

Step 6: Simplify and Convert

Simplify the fraction:
$\frac{4}{10} \times \frac{3}{9} \times \frac{2}{8} = \frac{4 \times 3 \times 2}{10 \times 9 \times 8} = \frac{24}{720} = \frac{1}{30} \approx 0.033$ (to thousandths).
Percentage: $0.033 \times 100 \approx 3.33\%$.

Step 1: Identify Event Type

Picking from two separate bags: the result from one bag does not affect the other. So, independent events.

Step 2: Probability of Blue (First Bag)

First bag: $4$ red + $5$ blue = $9$ marbles.
$P(\text{blue}) = \frac{5}{9}$.

Step 3: Probability of Yellow (Second Bag)

Second bag: $5$ green + $6$ yellow = $11$ marbles.
$P(\text{yellow}) = \frac{6}{11}$.

Step 4: Multiply for Joint Probability

For independent events, $P(A \cap B) = P(A) \times P(B)$.
So:
$$P = \frac{5}{9} \times \frac{6}{11}$$

Step 5: Simplify and Convert

Simplify: $\frac{5 \times 6}{9 \times 11} = \frac{30}{99} = \frac{10}{33} \approx 0.303$ (to thousandths).
Percentage: $0.303 \times 100 \approx 30.30\%$.

Answer:

• Event Type: Independent
• Probability (fraction): $\frac{1}{4}$
• Probability (decimal): $0.250$
• Probability (percentage): $25.00\%$

Now, problem 10: