We present combined $approx 14-37~rm ks$ Chandra observations of seven $z = 1.6-2.7$ broad absorption line (BAL) quasars selected from the Large Bright Quasar Survey (LBQS). These seven objects are high-ionization BAL (HiBAL) quasars, and they were undetected in the Chandra hard band ($2-8$ keV) in previous observations. The stacking analyses of previous Chandra observations suggested that these seven objects likely contain some candidates for intrinsically X-ray weak BAL quasars. With the new Chandra observations, six targets are detected. We calculate their effective power-law photon indices and hard-band flux weakness, and find that two objects, LBQS $1203+1530$ and LBQS $1442-0011$, show soft/steep spectral shapes ($Gamma_{rm eff}= 2.2^{+0.9}_{-0.9}$ and $1.9_{-0.8}^{+0.9}$) and significant X-ray weakness in the hard band (by factors of $approx$ 15 and 12). We conclude that the two HiBAL quasars are good candidates for intrinsically X-ray weak BAL quasars. The mid-infrared-to-UV spectral energy distributions (SEDs) of the two candidates are consistent with those of typical quasars. We constrain the fraction of intrinsically X-ray weak AGNs among HiBAL quasars to be $approx 7-10%$ ($2/29-3/29$), and we estimate it is $approx 6- 23%$ ($2/35-8/35$) among the general BAL quasar population. Such a fraction is considerably larger than the fraction among non-BAL quasars, and we suggest that intrinsically X-ray weak quasars are preferentially observed as BAL quasars. Intrinsically X-ray weak AGNs likely comprise a small minority of the luminous type 1 AGN population, and they should not affect significantly the completeness of these AGNs found in deep X-ray surveys.
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