Question

bryant is shopping for a new bicycle. he is most interested in color and type of tires.
road bike tires red 3 green 2
mountain bike tires red 3 green 8
city bike tires red 9 green 4
what is the probability that a randomly selected bike has mountain bike tires and is not red?
simplify any fractions.

Explanation:

Step1: Calculate total number of bikes

First, we sum up all the values in the table. The values are 3 (Road - Red), 2 (Road - Green), 3 (Mountain - Red), 8 (Mountain - Green), 9 (City - Red), 4 (City - Green). So total \( n = 3 + 2 + 3 + 8 + 9 + 4 \).
\( n=3 + 2+3 + 8+9 + 4=29 \)? Wait, no, wait: 3+2=5, 5+3=8, 8+8=16, 16+9=25, 25+4=29? Wait, no, let's recalculate: 3 (Road Red) + 2 (Road Green) = 5; 3 (Mountain Red) + 8 (Mountain Green) = 11; 9 (City Red) + 4 (City Green) = 13. Then total is 5 + 11 + 13 = 29? Wait, no, 5+11=16, 16+13=29. Wait, but let's check the number of bikes with mountain bike tires and not red: that's the green mountain bike tires, which is 8. Wait, no: "mountain bike tires and is not red" means mountain bike tires (regardless of color? No, "not red" so color is green. So the number of favorable outcomes is the number of mountain bike tires that are green, which is 8. Wait, no, the table: Mountain bike tires row, Green column is 8. Then total number of bikes is sum of all cells: 3 (Road Red) + 2 (Road Green) + 3 (Mountain Red) + 8 (Mountain Green) + 9 (City Red) + 4 (City Green) = 3+2=5, 5+3=8, 8+8=16, 16+9=25, 25+4=29? Wait, no, 3+2=5, 3+8=11, 9+4=13; 5+11=16, 16+13=29. So total number of bikes \( N = 3 + 2 + 3 + 8 + 9 + 4 = 29 \)? Wait, no, 3 (Road Red) + 2 (Road Green) = 5; 3 (Mountain Red) + 8 (Mountain Green) = 11; 9 (City Red) + 4 (City Green) = 13. 5 + 11 + 13 = 29. Then the number of bikes with mountain bike tires and not red is the number of mountain bike tires that are green, which is 8. Wait, but let's confirm: "mountain bike tires and is not red" – so the intersection of mountain bike tires and green (since not red is green here, as colors are red and green). So the cell for Mountain bike tires and Green is 8. So the number of favorable cases \( n = 8 \). Total cases \( N = 3 + 2 + 3 + 8 + 9 + 4 = 29 \)? Wait, no, 3+2=5, 3+8=11, 9+4=13; 5+11=16, 16+13=29. Wait, but let's check again: 3 (Road Red) + 2 (Road Green) = 5; 3 (Mountain Red) + 8 (Mountain Green) = 11; 9 (City Red) + 4 (City Green) = 13. 5 + 11 + 13 = 29. So probability is \( \frac{8}{29} \)? Wait, no, wait: maybe I miscalculated total. Let's add all numbers: 3 (Road Red) + 2 (Road Green) + 3 (Mountain Red) + 8 (Mountain Green) + 9 (City Red) + 4 (City Green) = 3+2=5, 5+3=8, 8+8=16, 16+9=25, 25+4=29. Yes. So the number of bikes with mountain bike tires and not red is 8 (Mountain Green). So probability is \( \frac{8}{29} \)? Wait, but let's check again. Wait, the problem says "mountain bike tires and is not red". So mountain bike tires: the row is Mountain bike tires. Not red: so column is Green (since the other color is Red). So the cell is 8. Total number of bikes: sum all cells. So 3+2+3+8+9+4=29. So probability is 8/29. Wait, but let's confirm the total again. 3 (Road Red) + 2 (Road Green) = 5; 3 (Mountain Red) + 8 (Mountain Green) = 11; 9 (City Red) + 4 (City Green) = 13. 5 + 11 = 16; 16 + 13 = 29. Correct. So the number of favorable is 8, total is 29. So probability is 8/29.

Step2: Identify favorable and total

Favorable: mountain bike tires (green) = 8. Total: sum of all = 29. So probability \( P = \frac{8}{29} \).

Answer:

\(\frac{8}{29}\)