Question

A player can take a maximum of 4 chances to hit a bottle with a flying disc.The probability of hitting the bottle at the first, second,third and fourth shots are 0.1, 0.2, 0.35 and 0.45 respectively.What is the probability that the player hits the bottle with the flying disc?

• A. 0.6573
• B. 0.2574
• C. 0.7426
• D. 0.3427

Answer 1

The Correct Option is C

To determine the probability that the player hits the bottle with the flying disc, we will use the concept of complementary probability. Complementary probability involves calculating the probability of the event not happening and then subtracting it from 1 to find the probability of the event happening. 

Let's denote the probability of hitting the bottle on the first, second, third, and fourth attempts as follows:

• \(P(A_1) = 0.1\) (Probability of hitting on the first shot)
• \(P(A_2) = 0.2\) (Probability of hitting on the second shot)
• \(P(A_3) = 0.35\) (Probability of hitting on the third shot)
• \(P(A_4) = 0.45\) (Probability of hitting on the fourth shot)

First, we calculate the probability of missing the bottle on each of the four shots:

• \(P(\overline{A_1}) = 1 - P(A_1) = 1 - 0.1 = 0.9\) (Probability of missing on the first shot)
• \(P(\overline{A_2}) = 1 - P(A_2) = 1 - 0.2 = 0.8\) (Probability of missing on the second shot)
• \(P(\overline{A_3}) = 1 - P(A_3) = 1 - 0.35 = 0.65\) (Probability of missing on the third shot)
• \(P(\overline{A_4}) = 1 - P(A_4) = 1 - 0.45 = 0.55\) (Probability of missing on the fourth shot)

The probability of missing the bottle in all four attempts is the product of individual probabilities of missing:

• \(P(\text{Missing all attempts}) = 0.9 \times 0.8 \times 0.65 \times 0.55\)
• Calculate this probability:
• \(P(\text{Missing all attempts}) = 0.9 \times 0.8 \times 0.65 \times 0.55 = 0.2574\)

Now, the probability of hitting the bottle in at least one of the four attempts is the complement of the probability of missing all attempts:

• \(P(\text{At least one hit}) = 1 - P(\text{Missing all attempts})\)
• \(P(\text{At least one hit}) = 1 - 0.2574 = 0.7426\)

Thus, the probability that the player hits the bottle with the flying disc is \(0.7426\).

Therefore, the correct answer is 0.7426.

Answer 2

The probability that the player does not hit the bottle on a given shot is:

• First shot: \( 1 - 0.1 = 0.9 \)
• Second shot: \( 1 - 0.2 = 0.8 \)
• Third shot: \( 1 - 0.35 = 0.65 \)
• Fourth shot: \( 1 - 0.45 = 0.55 \)

The probability that the player misses all 4 shots is:

\[ 0.9 \times 0.8 \times 0.65 \times 0.55 = 0.2874 \]

Thus, the probability that the player hits the bottle in at least one of the four shots is:

\[ 1 - 0.2874 = 0.7426 \]