Grandpa Hill has a bag of coins. The bag contains 5 silver dollars, 6 quarters, 2 dimes, and 2 nickels.

He is going to let his granddaughter Laynee randomly select and keep 2 coins from the bag one at a time without replacement.

What is the probability that Laynee will select a silver dollar both times?
Responses

Considering no replacement, and 2 silver coins must be chosen each time, the probability of our sample changes, and out calculations must adapt to it.

5 silv 4 silv (1 picked) 20

____ x __________ = ________

15 coins total 14 coins (no replac) 210

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Answer Link

semsee45 semsee45 · Answer 2 · 2024-03-23T08:26:36+01:00

Answer:

[tex]\sf \dfrac{2}{21}[/tex]

Step-by-step explanation:

Probability is the measure of the likelihood of an event occurring, and can be calculated by dividing the number of favourable outcomes by the total number of possible outcomes.

[tex]\large\boxed{\sf Probability=\dfrac{\textsf{Number of favourable outcomes}}{\textsf{Total number of possible outcomes}}}[/tex]

To find the probability that Laynee will select a silver dollar both times, we need to calculate the probability of selecting a silver dollar on the first draw and then the probability of selecting another silver dollar on the second draw.

[tex]\dotfill[/tex]

First Draw

The number of favourable outcomes for the first draw is the total number of silver dollars in the bag, which is 5.

The number of possible outcomes for the first draw is the total number of coins:

[tex]\sf 5 + 6 + 2 + 2 = 15[/tex]

Therefore, the probability that Laynee will select a silver dollar on the first draw is:

[tex]\sf P(S_1)=\dfrac{5}{15}[/tex]

[tex]\dotfill[/tex]

Second Draw

After the first draw, there are now 4 silver dollars left in the bag, and the total number of coins has decreased by one to 14, so the probability that Laynee will select a silver dollar on the second draw is:

[tex]\sf P(S_2)=\dfrac{4}{14}[/tex]

[tex]\dotfill[/tex]

Solution

To find the probability of both events happening (selecting a silver dollar both times), we multiply the probabilities of the individual events:

[tex]\sf P(\textsf{selecting a silver dollar both times}) = P(S_1) \times P(S_2) \\\\\\P(\textsf{selecting a silver dollar both times}) = \dfrac{5}{15} \times \dfrac{4}{14} \\\\\\P(\textsf{selecting a silver dollar both times}) = \dfrac{20}{210}\\\\\\P(\textsf{selecting a silver dollar both times}) = \dfrac{20\div 10}{210\div 10}\\\\\\P(\textsf{selecting a silver dollar both times}) = \dfrac{2}{21}[/tex]

Therefore, the probability that Laynee will select a silver dollar both times is:

[tex]\Large\boxed{\boxed{\sf \dfrac{2}{21}}}[/tex]