Question

A forecast is to be made of the results of five cricket matches, each of which an be a win or a draw or a loss for Indian team. Let,
$p=$ number of forecasts with exactly $1$ error
$q=$ number of forecasts with exactly $3$ errors and
$r=$ number of forecasts with all five errors
Then the correct statement(s) is/are

• A. $2q=5r$
• B. $8p=q$
• C. $8p=5r$
• D. $2(p+r)>q$

Answer 1

Answer: (A), (B), (D)

$2q=5r$
$2(p+r)>q$
$8p=q$

One error: First we select the match where the error occurred. Then we should keep in mind that an error can occur in 2 ways. If the outcome is say, win, then we can erroneously say loss or draw.
Thus, number of ways = ${}^5{C}_1\times 2=10$
Hence, $p=10$
Three errors: First we select the 3 matches where the error occurred. Then we should keep in mind that an error can occur in 2 ways, hence for 3 matches, errors can occur in 2*2*2 ways.
Thus, number of ways = ${}^5{C}_3\times 2^3=80$
Hence, $q=80$
For all 5 errors: the number of ways ${=}^5{C}_5\times 2^5=32$
Thus, $r=32$
Hence, (a), (b) and (d) are correct.