Questions: Determine whether the following individual events are independent or dependent. Then find the probability of the combined event. Drawing three tens in a row from a standard deck of cards when the drawn card is not returned to the deck each time The event of drawing a ten and the event of drawing a ten the next time are dependent. The probability of drawing three tens in a row from a standard deck of cards when the drawn card is not returned to the deck each time is (Type an integer or a simplified fraction.)

Solution Steps

The probability of drawing a ten on the first draw from a standard deck of cards is given by:

\[ P(\text{First Ten}) = \frac{4}{52} = \frac{1}{13} \approx 0.0769 \]

After drawing one ten, there are now 3 tens left in a deck of 51 cards. The probability of drawing a ten on the second draw is:

\[ P(\text{Second Ten}) = \frac{3}{51} = \frac{1}{17} \approx 0.0588 \]

After drawing two tens, there are 2 tens left in a deck of 50 cards. The probability of drawing a ten on the third draw is:

\[ P(\text{Third Ten}) = \frac{2}{50} = \frac{1}{25} = 0.04 \]

The combined probability of drawing three tens in a row is the product of the individual probabilities:

\[ P(\text{Three Tens}) = P(\text{First Ten}) \times P(\text{Second Ten}) \times P(\text{Third Ten}) = \frac{4}{52} \times \frac{3}{51} \times \frac{2}{50} \]

Calculating this gives:

\[ P(\text{Three Tens}) = \frac{4 \times 3 \times 2}{52 \times 51 \times 50} = \frac{24}{132600} = \frac{1}{5525} \approx 0.000181 \]

Final Answer