' The probability of winning a game is 1/10. You play 5 games. a)…

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Anu Bhardwaj · Accepted Answer

Step 1:- To solve these problems, we'll use the binomial probability formula, which calculates the probability of getting exactly 𝑘 k successes in 𝑛 n trials, given the probability of success 𝑝 p . The binomial probability formula is: 𝑃 ( 𝑋 = 𝑘 ) = ( 𝑛 𝑘 ) × 𝑝 𝑘 × ( 1 − 𝑝 ) 𝑛 − 𝑘 P ( X = k ) = ( k n ​ ) × p k × ( 1 − p ) n − k Explanation:- 𝑃 ( 𝑋 = 𝑘 ) P ( X = k ) is the probability of getting exactly 𝑘 k successes. 𝑛 n is the number of trials. 𝑘 k is the number of successes. 𝑝 p is the probability of success. ( 𝑛 𝑘 ) ( k n ​ ) is the number of combinations of choosing 𝑘 k successes out of 𝑛 n trials. Step 2:- 𝑝 = 1 10 p = 10 1 ​ (probability of winning a game) 𝑛 = 5 n = 5 (number of games) Let's solve each part: a) Probability of winning exactly 1 game: 𝑃 ( 𝑋 = 1 ) = ( 5 1 ) × ( 1 10 ) 1 × ( 9 10 ) 4 P ( X = 1 ) = ( 1 5 ​ ) × ( 10 1 ​ ) 1 × ( 10 9 ​ ) 4 = 5 × 1 10 × ( 9 10 ) 4 = 5 × 10 1 ​ × ( 10 9 ​ ) 4 ≈ 0.32805 =0.32805 b) Probability of winning all 5 games: 𝑃 ( 𝑋 = 5 ) = ( 5 5 ) × ( 1 10 ) 5 × ( 9 10 ) 0 P ( X = 5 ) = ( 5 5 ​ ) × ( 10 1 ​ ) 5 × ( 10 9 ​ ) 0 = 1 × ( 1 10 ) 5 × 1 = 1 × ( 10 1 ​ ) 5 × 1 ≈ 0.00001 =0.00001 c) Probability of winning 4 games in a row The problem requires us to find the probability of winning 4 consecutive games out of the 5 games played. To win 4 games in a row, it can happen in the following sequences: WWWWL (Wins in the first 4 games and loses the 5th game) LWWWW (Loses the first game and then wins the next 4 games) Let's calculate these probabilities correctly: WWWWL: The probability of winning the first 4 games and losing the 5th game is: 𝑃 (WWWWL)=(0.1)4×(0.9)1=0.14×0.9P(WWWWL)=(0.1)4×(0.9)1=0.14×0.9 =0.0001×0.9=0.00009=0.0001×0.9=0.00009 LWWWW: The probability of losing the first game and then winning the next 4 games is: 𝑃 (LWWWW)=(0.9)1×(0.1)4=0.9×0.14P(LWWWW)=(0.9)1×(0.1)4=0.9×0.14 =0.9×0.0001=0.00009=0.9×0.0001=0.00009 Now, summing these probabilities gives the total probability of winning 4 games in a row: 𝑃 (4 games in a row)= 𝑃 (WWWWL)+ 𝑃 (LWWWW)=0.00009+0.00009=0.00018P(4 games in a row)=P(WWWWL)+P(LWWWW)=0.00009+0.00009=0.00018 So, the correct probability of winning 4 games in a row is:  0.00018 d) Explanation:- The probability of winning exactly 1 game is 0.32805. The probability of winning all 5 games is 0.00001.  The probability of winning 4 games in a row is  0.00018.  won 4 games, what is the probability that you have won 4 in a row: 0.40. Final Answer:- a) Probability of winning exactly 1 game: 0.32805 b) Probability of winning all 5 games: 0.00001 c) Probability of winning 4 games in a row:    0.00018 d) won 4 games, what is the probability that you have won 4 in a row: 0.40

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Statistics and probability

Anu Bhardwaj

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