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A bag contains 5 red marbles, 4 blue marbles and 7 green marbles. If three marbles are drawn out of the bag, what is the exact probability that all three marbles drawn will be red? Answer:

A bag contains 55 red marbles, 44 blue marbles and 77 green marbles. If three marbles are drawn out of the bag, what is the exact probability that all three marbles drawn will be red?\newlineAnswer:

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Q. A bag contains 55 red marbles, 44 blue marbles and 77 green marbles. If three marbles are drawn out of the bag, what is the exact probability that all three marbles drawn will be red?\newlineAnswer:
  1. Determine total number of marbles: Determine the total number of marbles in the bag.\newlineThe bag contains 55 red marbles, 44 blue marbles, and 77 green marbles. So, the total number of marbles is 5+4+7=165 + 4 + 7 = 16 marbles.
  2. Calculate probability of drawing one red marble: Calculate the probability of drawing one red marble. The probability of drawing one red marble is the number of red marbles divided by the total number of marbles, which is 516\frac{5}{16}.
  3. Calculate probability of drawing second red marble: Calculate the probability of drawing a second red marble after one has already been drawn.\newlineAfter drawing one red marble, there are now 44 red marbles left and 1515 marbles in total. The probability of drawing a second red marble is now 415\frac{4}{15}.
  4. Calculate probability of drawing third red marble: Calculate the probability of drawing a third red marble after two have already been drawn.\newlineAfter drawing two red marbles, there are now 33 red marbles left and 1414 marbles in total. The probability of drawing a third red marble is now 314\frac{3}{14}.
  5. Calculate probability of all three events: Calculate the probability of all three events happening in sequence (drawing three red marbles in a row).\newlineThe probability of all three events happening in sequence is the product of the probabilities of each event. This is calculated as (516)×(415)×(314)(\frac{5}{16}) \times (\frac{4}{15}) \times (\frac{3}{14}).
  6. Perform multiplication to find exact probability: Perform the multiplication to find the exact probability.\newlineThe exact probability is (516)×(415)×(314)=(5×4×316×15×14)=603360(\frac{5}{16}) \times (\frac{4}{15}) \times (\frac{3}{14}) = (\frac{5 \times 4 \times 3}{16 \times 15 \times 14}) = \frac{60}{3360}.
  7. Simplify fraction to lowest terms: Simplify the fraction to its lowest terms.\newlineThe fraction 603360\frac{60}{3360} can be simplified by dividing both the numerator and the denominator by the greatest common divisor, which is 6060. So, 603360\frac{60}{3360} simplifies to 156\frac{1}{56}.

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