When a particular type of thumbtack is dropped, it will land point up ( \( \perp \) ) or point down ( K ). This experiment was repeated 90 times with the following results: point up: 58 times; point down: 32 times. a. What is the experimental probability that a particular type of thumbtack will land point up? b. What is the experimental probability that a particular type of thumbtack will land point down? c. If the experiment was tried another 90 times would the same results occur? Why? d. Is it expected that nearly the same results occur on a second trial? Why? a. The experimental probability that a particular type of thumbtack will land point up is (Type an integer or a simplified fraction.)

The experimental probability that a particular type of thumbtack will land point up is calculated by dividing the number of times it landed point up by the total number of trials. So, that's 58 times landing point up out of 90 total trials, which gives us \( \frac{58}{90} \) or simplified, \( \frac{29}{45} \). The experimental probability that a particular type of thumbtack will land point down is found similarly. You take the number of times it landed point down (32) and divide that by the total number of trials (90), resulting in \( \frac{32}{90} \) which simplifies to \( \frac{16}{45} \). If you repeat the experiment another 90 times, there’s a chance the results may vary. Random events, like dropping thumbtacks, can produce different outcomes each time due to factors like slight variations in the way they are dropped or even minor differences in the thumbtacks themselves. In theory, while you might expect a similar ratio of outcomes, it's absolutely possible that the results won't match exactly due to randomness. Probability gives us a trend over many trials, but it can't guarantee a specific outcome every time. That's part of the fun with statistics – roll the dice and let's see what happens!