Contractor X is developing his bidding strategy against Contractor Y. The ratio of Y's bid price to X's cost for the 30 previous bids in which Contractor X has competed against Contractor Y is given in the Table

Ratio of Y's bid price to X's cost	Number of bids
1.02	6
1.04	12
1.06	3
1.10	6
1.12	3

Based on the bidding behaviour of the Contractor Y, the probability of winning against Contractor Y at a mark up of 8% for the next project is

Contractor X Bidding Strategy Analysis

This solution calculates the probability of Contractor X winning a bid against Contractor Y, using historical bid data and a specific markup strategy.

Historical Bid Data Summary

The table provides the frequency distribution of the ratio $\frac{\text{Y’s Bid Price}}{\text{X’s Cost}}$ for 30 previous bids:The total number of bids analyzed is $6+12+3+6+3=30$.

Determining the Winning Condition

Contractor X plans to use an 8% markup. Contractor X's bid price is calculated as:

${\text{Bid Price}}_X={\text{Cost}}_X\times (1+0.08)=1.08\times {\text{Cost}}_X$

Contractor X wins if their bid price is less than Contractor Y's bid price:

${\text{Bid Price}}_X<{\text{Bid Price}}_Y$

Substituting the formula for X's bid price:

$1.08\times {\text{Cost}}_X<{\text{Bid Price}}_Y$

To determine the winning condition based on the provided ratio, we rearrange the inequality:

$1.08<\frac{{\text{Bid Price}}_Y}{{\text{Cost}}_X}$

This means Contractor X wins if the ratio of Y's bid price to X's cost is greater than 1.08.

Probability Calculation Steps

We count the historical bids where the ratio was greater than 1.08:

- Ratio = 1.10: 6 bids
- Ratio = 1.12: 3 bids

The total number of bids where Contractor X would have won is $6+3=9$.

The probability of winning is the number of winning bids divided by the total number of bids:

$\text{P(Win)}=\frac{\text{Number of bids with Ratio}>1.08}{\text{Total Number of Bids}}$

$\text{P(Win)}=\frac{9}{30}$

$\text{P(Win)}=0.3$ or 30%

Result and Conclusion

The calculated probability of Contractor X winning the next bid with an 8% markup is 30%.

Comparing this probability to the given options:

- 0%
- more than 0% but less than 50%
- more than 50% but less than 100%
- 100%

Since 30% is greater than 0% and less than 50%, Option 2 is the correct range.