Question

a gymnastics coach records the scores of a gymnast and develops the probability distribution below. what is the probability the gymnast scores a 9 or 9.5? probability distribution gymnastics score: x probability: p(x) 5 0.02 5.5 0.02 6 0.06 6.5 0.1 7 0.16 7.5 0.14 8 0.18 8.5 0.18 9 ? 9.5 0.02

Explanation:

Step1: Find total of known probabilities

Sum the given probabilities: \(0.02 + 0.02 + 0.06 + 0.1 + 0.16 + 0.14 + 0.18 + 0.18 + 0.02\)
\(= 0.02\times3 + 0.06 + 0.1 + 0.16 + 0.14 + 0.18\times2\)
\(= 0.06 + 0.06 + 0.1 + 0.16 + 0.14 + 0.36\)
\(= 0.06+0.06=0.12\); \(0.12+0.1=0.22\); \(0.22+0.16=0.38\); \(0.38+0.14=0.52\); \(0.52+0.36=0.88\)

Step2: Calculate \(P(9)\)

Since total probability is \(1\), \(P(9) = 1 - 0.88 = 0.12\)? Wait, no—wait, the known probabilities: let's recalculate. Wait, the scores are 5,5.5,6,6.5,7,7.5,8,8.5,9,9.5. Wait, I missed 9? No, the table has 5,5.5,6,6.5,7,7.5,8,8.5,9,9.5. Wait, the given probabilities: 0.02 (5), 0.02 (5.5), 0.06 (6), 0.1 (6.5), 0.16 (7), 0.14 (7.5), 0.18 (8), 0.18 (8.5), 0.02 (9.5). Wait, I missed 9? Wait, no, the 9 row is "?", and 9.5 is 0.02. Let's sum again:

\(0.02 + 0.02 = 0.04\); \(+0.06 = 0.1\); \(+0.1 = 0.2\); \(+0.16 = 0.36\); \(+0.14 = 0.5\); \(+0.18 = 0.68\); \(+0.18 = 0.86\); \(+0.02 = 0.88\). So total known (excluding 9) is 0.88. Thus \(P(9) = 1 - 0.88 = 0.12\)? Wait, no—wait, 5,5.5,6,6.5,7,7.5,8,8.5,9,9.5: that's 10 scores. Wait, 5 (1), 5.5 (2), 6 (3), 6.5 (4), 7 (5), 7.5 (6), 8 (7), 8.5 (8), 9 (9), 9.5 (10). So the sum of all \(P(X)\) must be 1. So sum of the 9 given (excluding 9) is 0.88, so \(P(9) = 1 - 0.88 = 0.12\)? Wait, no, wait the 9.5 is 0.02, so the 9th row (9) is missing. Wait, let's list all:

• 5: 0.02
• 5.5: 0.02
• 6: 0.06
• 6.5: 0.1
• 7: 0.16
• 7.5: 0.14
• 8: 0.18
• 8.5: 0.18
• 9:?
• 9.5: 0.02

Sum these: \(0.02 + 0.02 = 0.04\); \(+0.06 = 0.1\); \(+0.1 = 0.2\); \(+0.16 = 0.36\); \(+0.14 = 0.5\); \(+0.18 = 0.68\); \(+0.18 = 0.86\); \(+0.02 = 0.88\). So \(P(9) = 1 - 0.88 = 0.12\).

Step3: Find \(P(9 \text{ or } 9.5)\)

Use the addition rule for mutually exclusive events: \(P(9 \cup 9.5) = P(9) + P(9.5)\)
We have \(P(9) = 0.12\) (from Step2) and \(P(9.5) = 0.02\).
So \(0.12 + 0.02 = 0.14\)? Wait, no—wait, did I miscalculate \(P(9)\)? Wait, let's check again. Wait, the scores are 5,5.5,6,6.5,7,7.5,8,8.5,9,9.5: 10 scores. The sum of all probabilities must be 1. Let's sum the given (excluding 9):

0.02 (5) + 0.02 (5.5) = 0.04
+0.06 (6) = 0.1
+0.1 (6.5) = 0.2
+0.16 (7) = 0.36
+0.14 (7.5) = 0.5
+0.18 (8) = 0.68
+0.18 (8.5) = 0.86
+0.02 (9.5) = 0.88

So \(P(9) = 1 - 0.88 = 0.12\). Then \(P(9 \text{ or } 9.5) = 0.12 + 0.02 = 0.14\)? Wait, but let's check again. Wait, maybe I made a mistake in the number of terms. Wait, 5,5.5,6,6.5,7,7.5,8,8.5,9,9.5: that's 10 rows. The given probabilities (excluding 9) are 9 rows? No, 5,5.5,6,6.5,7,7.5,8,8.5,9.5: that's 9 rows, plus 9: 10. So sum of 9 rows (excluding 9) is 0.88, so 9's probability is 0.12. Then 9 or 9.5 is 0.12 + 0.02 = 0.14. Wait, but let's verify the sum again. Wait, 0.02+0.02=0.04; +0.06=0.1; +0.1=0.2; +0.16=0.36; +0.14=0.5; +0.18=0.68; +0.18=0.86; +0.02=0.88. Yes. So 1 - 0.88 = 0.12. Then 0.12 + 0.02 = 0.14.

Answer:

\(0.14\)